This website uses cookies to ensure you get the best experience. set of all real numbers. We can extend the line to the left and right by repeating, and then draw a line through the points. Graphs of linear functions may be transformed by shifting the graph up, down, left, or right as well as using stretches, compressions, and reflections. In addition, the graph has a downward slant which indicates a negative slope. Graphing Linear Equations Find the Equation of a Line. Now we have to determine the slope of the line. No. This graph illustrates vertical shifts of the function [latex]f\left(x\right)=x[/latex]. In general, a linear function28 is a function that can be written in the form f(x) = mx + b LinearFunction where the slope m and b represent any real numbers. Test. The graph below is of the function [latex]f\left(x\right)=-\frac{2}{3}x+5[/latex]. This function includes a fraction with a denominator of 3 so let’s choose multiples of 3 as input values. By graphing two functions, then, we can more easily compare their characteristics. First, graph y = x. Graphing a Linear Equation by Plotting Three Ordered Pairs. Keep in mind that a vertical line is the only line that is not a function.). The a represents the gradient of the line, which gives the rate of change of the dependent variable. Book From our example, we have [latex]m=\frac{1}{2}[/latex], which means that the rise is 1 and the run is 2. Relating linear contexts to graph features Get 5 of 7 questions to level up! So, for this definition, the above function is linear only when c = 0, that is when the line passes through the origin. The steepness of a hill is called a slope. This means the larger the absolute value of m, the steeper the slope. The equation for the function also shows that [latex]b=-3[/latex], so the identity function is vertically shifted down 3 units. Key Concepts: Terms in this set (10) Which values of m and b will create a system of equations with no solution? This tells us that for each vertical decrease in the “rise” of [latex]–2[/latex] units, the “run” increases by 3 units in the horizontal direction. In general, a linear function Any function that can be written in the form f ( x ) = m x + b is a function that can be written in the form f ( x ) = m x + b L i n e a r F u n c t i o n where the slope m and b represent any real … f(a) is called a function, where a … From the initial value (0, 5) we move down 2 units and to the right 3 units. Graph linear functions. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. You need only two points to graph a linear function. By graphing two functions, then, we can more easily compare their characteristics. -x + y = 3. Free linear equation calculator - solve linear equations step-by-step. BYJU’S online graphing linear equations calculator tool makes the calculation faster and it displays the graph in a fraction of seconds. Graph Linear Equations using Slope-Intercept We can use the slope and y-intercept to graph a linear equation. The third is applying transformations to the identity function [latex]f\left(x\right)=x[/latex]. Choosing three points is often advisable because if all three points do not fall on the same line, we know we made an error. GeoGebra Classroom Activities. ; b = where the line intersects the y-axis. Solving Systems of Linear Equations: Graphing. We were also able to see the points of the function as well as the initial value from a graph. Furthermore, the domain and range consists of all real numbers. STUDY. The y-intercept and slope of a line may be used to write the equation of a line. All linear functions cross the y-axis and therefore have y-intercepts. Linear Function Graph. Note: A function f (x) = b, where b is a constant real number is called a constant function. Select two options. The y-intercept is the point on the graph when x = 0. The order of the transformations follows the order of operations. Often, the number in front of x is already a fraction, so you won't have to convert it. There are three basic methods of graphing linear functions. Recall that the slope is the rate of change of the function. Did you have an idea for improving this content? We were also able to see the points of the function as well as the initial value from a graph. Previously, we saw that that the graph of a linear function is a straight line. Show Step-by-step Solutions. When m is negative, there is also a vertical reflection of the graph. For distinguishing such a linear function from the other concept, the term affine function is often used. Each graphing linear equations worksheet on this page has four coordinate planes and equations in slope-intercept form, and includes an answer key showing the correct graph. Find a point on the graph we drew in Example: Graphing by Using the y-intercept and Slope that has a negative x-value. In this non-linear system, users are free to take whatever path through the material best serves their needs. [latex]\begin{array}{llllll}x=0& & f\left(0\right)=-\frac{2}{3}\left(0\right)+5=5\Rightarrow \left(0,5\right)\\ x=3& & f\left(3\right)=-\frac{2}{3}\left(3\right)+5=3\Rightarrow \left(3,3\right)\\ x=6& & f\left(6\right)=-\frac{2}{3}\left(6\right)+5=1\Rightarrow \left(6,1\right)\end{array}[/latex]. Its graph is a horizontal line at y = b. Begin by choosing input values. Linear functions are functions that produce a straight line graph.. The first characteristic is its y-intercept which is the point at which the input value is zero. Another option for graphing is to use transformations on the identity function [latex]f\left(x\right)=x[/latex]. Interpret solutions to linear equations and inequalities graphically. Linear equations word problems: graphs Get 3 of 4 questions to level up! A y-intercept is a y-value at which a graph crosses the y-axis. Do all linear functions have y-intercepts? http://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@5.2, Graph a linear function by plotting points, Graph a linear function using the slope and y-intercept, Graph a linear function using transformations. Properties. Learn More at mathantics.comVisit http://www.mathantics.com for more Free math videos and additional subscription based content! This website uses cookies to ensure you get the best experience. Explore math with our beautiful, free online graphing calculator. Subtract x from each side. A graphing calculator can be used to verify that your answers "make sense" or "look right". A table of values might look as below. For example, following order of operations, let the input be 2. Functions: Hull: First graph: f(x) Derivative Integral From ... Mark points at: First graph: x= Second graph: x= Third graph: x= Reticule lines Axis lines Caption Dashes Frame Errors: Def. dillinghamt. By graphing two functions, then, we can more easily compare their characteristics. eval(ez_write_tag([[728,90],'analyzemath_com-medrectangle-3','ezslot_4',320,'0','0'])); Determine the x intercept, set f(x) = 0 and What is the slope of a linear function? After studying this section, you will be able to: 1. The. This is also known as the “slope.” The b represents the y-axis intercept. We were also able to see the points of the function as well as the initial value from a graph. A linear function is a function which forms a straight line in a graph. An x-intercept is an x-value at which a graph crosses the x-axis. The slope is defined as the ratio of the vertical change between two points, the rise, to the horizontal change between the same two points, the run. (See Getting Help in Stage 1.) Graph [latex]f\left(x\right)=-\frac{2}{3}x+5[/latex] by plotting points. Usage To plot a function just type it into the function box. Draw Function Graphs Mathematics / Analysis - Plotter - Calculator 4.0. The equation, written in this way, is called the slope-intercept form. Graph a straight line by finding its x - and y-intercepts. Graph Linear Equations in Two Variables Learning Objectives. Using slope and intercepts in context Get 3 of 4 questions to level up! The independent variable is x and the dependent variable is y. a is the constant term or the y intercept. Because y = f(x), we can use y and f(x) interchangeably, and ordered pair solutions on the graph (x, y) can be written in the form (x, f(x)). eval(ez_write_tag([[728,90],'analyzemath_com-medrectangle-4','ezslot_5',344,'0','0'])); Any function of the form Linear functions are those whose graph is a straight line. 1. If variable x is a constant x=c, that will represent a line paralel to y-axis. Graph a straight line by finding three ordered pairs that are solutions to the linear equation. The graph of a linear function is a line. Tell whether each function is linear. Possible answers include [latex]\left(-3,7\right)[/latex], [latex]\left(-6,9\right)[/latex], or [latex]\left(-9,11\right)[/latex]. The slope of a linear function is equal to the ratio of the change in outputs to the change in inputs. Linear functions are functions that produce a straight line graph. We can begin graphing by plotting the point (0, 1) We know that the slope is rise over run, [latex]m=\frac{\text{rise}}{\text{run}}[/latex]. Introduction to Linear Relationships: IM 8.3.5. The equation can be written in standard form, so the function is linear. Graphing Linear Functions. Identify and graph a linear function using the slope and y-intercept. Another way to graph linear functions is by using specific characteristics of the function rather than plotting points. A linear function has the following form. The graph of a linear relation can be found by plotting at least two points. For the given x-coordinates, find f(x) and complete the function tables. Evaluating the function for an input value of 1 yields an output value of 2 which is represented by the point (1, 2). These unique features make Virtual Nerd a viable alternative to private tutoring. In Activity 1 the learners should enter the expressions one by one into the graphing calculator and classify the functions according to the shape of the graph. Created by. The equation for a linear function is: y = mx + b, Where: m = the slope ,; x = the input variable (the “x” always has an exponent of 1, so these functions are always first degree polynomial.). Free functions and graphing calculator - analyze and graph line equations and functions step-by-step. Evaluate the function at each input value. This is also expected from the negative constant rate of change in the equation for the function. The graph crosses the y-axis at (0, 1). Examples: 1. Learn more Accept. Another way to graph linear functions is by using specific characteristics of the function rather than plotting points. Two competing telephone companies offer different payment plans. Recognize the standard form of a linear function. Dritter Graph: h(x) Ableitung Integral +C: Blau 1 Blau 2 Blau 3 Blau 4 Blau 5 Blau 6 Rot 1 Rot 2 Rot 3 Rot 4 Gelb 1 Gelb 2 Grün 1 Grün 2 Grün 3 Grün 4 Grün 5 Grün 6 Schwarz Grau 1 Grau 2 Grau 3 Grau 4 Weiß Orange Türkis Violett 1 Violett 2 Violett 3 Violett 4 Violett 5 Violett 6 Violett 7 Lila Braun 1 Braun 2 Braun 3 Zyan Transp. Find the slopes and the x- and y-intercepts of the following lines. Notice that adding a value of b to the equation of [latex]f\left(x\right)=x[/latex] shifts the graph of f a total of b units up if b is positive and |b| units down if b is negative. We repeat until we have multiple points, and then we draw a line through the points as shown below. Method 1: Graphing Linear Functions in Standard Form 1. f(0). Linear equations word problems: tables Get 3 of 4 questions to level up! Graph [latex]f\left(x\right)=\frac{1}{2}x - 3[/latex] using transformations. We will choose 0, 3, and 6. Using vertical stretches or compressions along with vertical shifts is another way to look at identifying different types of linear functions. The slope of a linear function will be the same between any two points. In the equation [latex]f\left(x\right)=mx[/latex], the m is acting as the vertical stretch or compression of the identity function. Evaluate when . We encountered both the y-intercept and the slope in Linear Functions. This is also known as the “slope.” The b represents the y-axis intercept. We can now graph the function by first plotting the y-intercept. Horizontal lines are written in the form, [latex]f(x)=b[/latex]. In mathematics, a graphing linear equation represents the graph of the linear equation. The graph of the linear equation will always result in a straight line. To draw the graph we need coordinates. Starting from our y-intercept (0, 1), we can rise 1 and then run 2 or run 2 and then rise 1. of f is the how to graph linear equations using the slope and y-intercept. It looks like the y-intercept (b) of the graph is 2, as represented by point (0,2). Linear functions are typically written in the form f(x) = ax + b. Learn. Furthermore, the domain and range consists of all real numbers. We then plot the coordinate pairs on a grid. You can move the graph of a linear function around the coordinate grid using transformations. The first characteristic is its y-intercept which is the point at which the input value is zero. To find the y-intercept, we can set [latex]x=0[/latex] in the equation. Students learn that the linear equation y = x, or the diagonal line that passes through the origin, is called the parent graph for the family of linear equations. These points may be chosen as the x and y intercepts of the graph for example. 8 Linear Equations Worksheets. Graph horizontal and vertical lines. 2. The input values and corresponding output values form coordinate pairs. To zoom, use the zoom slider. Another way to graph linear functions is by using specific characteristics of the function rather than plotting points. When we’re comparing two lines, if their slopes are equal they are parallel, and if they are in … Graph 3x - 2y = 8. How to transform linear functions, Horizontal shift, Vertical shift, Stretch, Compressions, Reflection, How do stretches and compressions change the slope of a linear function, Rules for Transformation of Linear Functions, PreCalculus, with video lessons, examples and step-by-step solutions. First, graph the identity function, and show the vertical compression. Interactive, free online graphing calculator from GeoGebra: graph functions, plot data, drag sliders, and much more! This topic covers: - Intercepts of linear equations/functions - Slope of linear equations/functions - Slope-intercept, point-slope, & standard forms - Graphing linear equations/functions - Writing linear equations/functions - Interpreting linear equations/functions - Linear equations/functions … The simplest way is to find the intercept values for both the x-axis and the y-axis. The first is by plotting points and then drawing a line through the points. Linear functions word problem: fuel (Opens a modal) Practice. Write the equation in standard form. Method 1: Graphing Linear Functions in Standard Form 1. These pdf worksheets provide ample practice in plotting the graph of linear functions. We know that the linear equation is defined as an algebraic equation in which each term should have an exponents value of 1. Use the resulting output values to identify coordinate pairs. Free functions and graphing calculator - analyze and graph line equations and functions step-by-step This website uses cookies to ensure you get the best experience. Graph [latex]f\left(x\right)=4+2x[/latex], using transformations. Flashcards. Plot the coordinate pairs and draw a line through the points. Then just draw a line that passes through both of these points. Recognize the standard form of a linear function. A similar word to linear function is linear correlation. Scroll down the page for more examples and solutions. Twitter. linear functions by the shape of their graphs and by noting differences in their expressions. Graphing Linear Functions. Graphing Linear Equations. To graph, choose three values of x, and use them to generate ordered pairs. The slopes in level 1 worksheets are in the form of integers. The Slider Area. The functions whose graph is a line are generally called linear functions in the context of calculus. This inequality notation means that we should plot the graph for values of x between and including -3 and 3. The slope-intercept form gives you the y- intercept at (0, –2). Solve a system of linear equations. f (x) = m x + b, where m is not equal to 0 is called a linear function. The slope of a line is a number that describes steepnessand direction of the line. Write. Because the given function is a linear function, you can graph it by using slope-intercept form. For example, Plot the graph of y = 2x – 1 for -3 ≤ x ≤ 3. Graphing Linear Equations Calculator is a free online tool that displays the graph of the given linear equation. How many solutions does this linear system have? According to the equation for the function, the slope of the line is [latex]-\frac{2}{3}[/latex]. When it comes to graphing linear equations, there are a few simple ways to do it. Graph Linear Equations by Plotting Points It takes only 2 points to draw a graph of a straight line. What is a Linear Function? Vertical stretches and compressions and reflections on the function [latex]f\left(x\right)=x[/latex]. And here is its graph: It makes a 45° (its slope is 1) It is called "Identity" because what comes out is identical to what goes in: In. Students also learn the different types of transformations of the linear parent graph. The graph of a linear function is always a line. Google+. How to Use this Applet Definitions +drag: Hold down the key, then drag the described object. b. Solver to Analyze and Graph a Linear Function. Knowing an ordered pair written in function notation is necessary too. Given a real-world situation that can be modeled by a linear function or a graph of a linear function, the student will determine and represent the reasonable domain and range of the linear function … b = where the line intersects the y-axis. Notice that multiplying the equation [latex]f\left(x\right)=x[/latex] by m stretches the graph of f by a factor of m units if m > 1 and compresses the graph of f by a factor of m units if 0 < m < 1. Regardless of whether a table is given to you, you should consider using one to ensure you’re correctly graphing linear and quadratic functions. Vertically stretch or compress the graph by a factor. m = -2 and b = -1/3 m = -2 and b = -2/3. The graph of this function is a line with slope − and y-intercept −. Although the linear functions are also represented in terms of calculus as well as linear algebra. Graphing a Linear Function Using y-intercept and Slope. When the function is evaluated at a given input, the corresponding output is calculated by following the order of operations. Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. +drag: Hold down the key, then drag the described object. Is the Function Linear or Nonlinear | Table. The following diagrams show the different methods to graph a linear equation. It has the unique feature that you can save your work as a URL (website link). 3. The, of this function is the set of all real numbers. y = f(x) = a + bx. However, in linear algebra, a linear function is a function that maps a sum to the sum of the images of the summands. Linear functions are those whose graph is a straight line. We previously saw that that the graph of a linear function is a straight line. The graph of a linear function is a straight line, while the graph of a nonlinear function is a curve. (Note: A vertical line parallel to the y-axis does not have a y-intercept. A Review of Graphing Lines. Determine the y intercept, set x = 0 to find Given the equations of two lines, determine whether their graphs are parallel or perpendicular. A linear equation is drawn as a straight line on a set of axes. The graph of a linear function is a line. Graphing Linear Function: Type 2 - Level 1. Evaluate the function at each input value and use the output value to identify coordinate pairs. Graphing linear functions (2.0 MiB, 1,144 hits) Slope Determine slope in slope-intercept form (160.4 KiB, 766 hits) Determine slope from given graph (2.1 MiB, 834 hits) Find the integer of unknown coordinate (273.6 KiB, 858 hits) Find the fraction of unknown coordinate (418.5 KiB, 891 hits) Linear inequalities Graph of linear inequality (2.8 MiB, 929 hits) Facebook. Examine the input(x) and output(y) values of the table inthese linear function worksheets for grade 8. In Linear Functions, we saw that that the graph of a linear function is a straight line. However, in linear algebra, a linear function is a function that maps a sum to the sum of the images of the summands. Write the equation for a linear function from the graph of a line. Evaluating the function for an input value of 2 yields an output value of 4 which is represented by the point (2, 4). Plot the points and graph the linear function. The same goes for the steepness of a line. In linear algebra, mathematical analysis, and functional analysis, a linear function is a … Draw a line which passes through the points. How to graph Linear Functions by finding the X-Intercept and Y-Intercept of the Function? PLAY. ++drag: Hold down both the key and the key, then drag the described object. The first … Two points that are especially useful for sketching the graph of a line are found with the intercepts. In this video we look at graphing equations using a table of values To find points of a function, we can choose input values, evaluate the function at these input values, and calculate output values. The negative constant rate of change in outputs to the right 3.! As the initial value from a graph the following lines two lines, determine whether graphs... Constant x=c, that will represent a line as expected a straight line graph point 0,2... Linear equations step-by-step plot a function may also be transformed by a factor same between linear function graph. So you wo n't have to find the at y = mx + b y = f ( ). Graph of the function as well as the initial value ( 0, 3, and more determine y. Two points intercept b. Solver to Analyze and graph a linear function will be able to see the points the... And nonlinear functions by finding three ordered pairs that are solutions to the change outputs! Equations word problems: graphs Get 3 of 4 questions to level up simplest! We will choose 0, –2 ) these unique features make Virtual Nerd viable. Which a graph of a linear function: a vertical line parallel perpendicular... Set x = 0 to find the y intercept, set x = 0 to find the y intercept set... ) =-\frac { 2 } [ /latex ] using transformations word problems: graphs Get 3 4! Hill is called a function f ( 0, 1 ) as an algebraic equation in each... The other concept, the steeper the slope is [ latex ] f\left ( x\right ) {! That a vertical line linear function graph indicates that this graph illustrates vertical shifts of the table inthese linear function a. Featured graphing Utility that supports graphing two functions together y-intercept of the line and.! 1 } { 2 } { 2 } [ /latex ] by plotting points + 3/2 ensure you the... The “ slope. ” the b represents the gradient of the function as well as initial... As well as the “ slope. ” the b represents the gradient the. /Latex ] domain and range consists of all real numbers points to graph a linear function is at... Equation and its visual representation the larger the absolute value of m, is! A straight line in a straight line graph being able to: 1, that represent. And involves using the slope is the set of all real numbers another option for graphing is find... Functions that produce a straight line graph by reversing the order of operations b -2/3... Graphs Mathematics / Analysis - linear function graph - calculator 4.0 mathantics.comVisit http: //www.mathantics.com for more examples detailed... And output ( y ) values of x, and show the different methods to,! Another option for graphing is to use this Applet Definitions < SHIFT > +drag: down... ≤ x ≤ 3 function graphs Mathematics / Analysis - Plotter - calculator 4.0 to plot a function Type... ) is called a constant x=c, that will represent a line may be used write. The slope-intercept method and involves using the slope is [ latex ] f\left ( x\right ) =-\frac { }! Are represented as fractions in the form f ( x ) = ax + b y b... At mathantics.comVisit http: //www.mathantics.com for more free math videos and additional subscription based content by following order. Save your work as a straight line to generate ordered pairs that are especially useful for sketching graph... When m is negative linear function graph there are three basic methods of graphing linear calculator. Coordinates by substituting values into the function equation in which each term should have an value. You have difficulties with this material, please contact your instructor and graphing calculator - Analyze and graph the functions! Level up this content so the function [ latex ] f\left ( x\right ) =4+2x [ /latex ] by points. Including -3 and 3 in Standard form ( a ) is called slope-intercept. Means that we should plot the graph of the given linear equation the y-intercept slope! Constant x=c, that will represent a line parallel or perpendicular word problem: fuel ( Opens a )... ) of the dependent variable noting differences in their expressions other concept, the has... Line as expected that that the slope is [ latex ] f\left ( x\right ) =\frac { 1 } 2... 4 questions to level up is not a function just Type it the! = b, where b is a line are found with the intercepts therefore have y-intercepts coordinate! Each term should have an idea for improving this content this function includes a fraction, so function. ( 0,2 ) more easily compare their characteristics Plotter - calculator 4.0 y-axis intercept difficulties with this material please! Private tutoring b is a curve –2 ) the intercept values for both y-intercept... Feature that you can graph it by using slope-intercept form at x = 0 is 5, so wo... Equation of a hill is called a constant function. ) multiples of 3 as input values can now the... Are generally called linear functions are functions that produce a straight line f\left! Vertical shifts is another way to graph this Type of function, b! Form gives you the y- intercept at ( 0, 1 ) two functions,,! By simply placing the value of 1 then plot the coordinate pairs and draw a line always! = a + bx extend the line intersects the y-axis at ( 0, –2 ) function whose degree utmost... Straight line a special linear function is the point at which the input ( x ) and complete the as. ) practice vertical shifts of the line methods of graphing linear equations.!, could linear function graph have multiple points, and then draw a line with slope m y... These coordinates by substituting values into the linear equation calculus as well linear. Compress the graph of a line are generally called linear functions are those whose is! In plotting the graph of a linear function from the graph of straight... Function: a function which forms a straight line the a represents the gradient of the in... To linear function. ) of the linear function: Type 1 - 1... Subscription based content is also known as the “ slope. ” the b the! Are in the equation of a line function which forms a straight line by finding the is. Take whatever path through the points number in front of x and the slope of a linear function Type! Vertically stretch or compress the graph of the dependent variable calculated by following the order operations. Step by step tutorial with examples and solutions ( x ) = +... Domain and range consists of all real numbers ax + b y = -2x + 3/2 equation in each! Up, down, left, or compression at least two points 1 worksheets are the... Its y-intercept which is a number that describes steepnessand direction of the function is a number describes. Steeper the slope of a linear function around the coordinate grid using transformations operations let. Material, please contact your instructor steepness of a line with slope m and y intercepts of the in! Same goes for the given x-coordinates, find f ( x ) -2x + 3/2 distinguishing such a linear will! By graphing two functions, we know the graph graph we drew in:! ) practice function is a line through the points and graph a straight line initial from! Students will have to determine the y intercept b. Solver to Analyze and graph a linear function: Type -. Table inthese linear function called the `` identity function, it is still important to practice each.... By observing the graphs to generate ordered pairs + 4 Solution to example 1 utmost 1 or linear function graph of... Equations step-by-step is necessary linear function graph a horizontal line at y = b, where a … function!, determine whether their graphs and by noting differences in their expressions a polynomial function degree. Points to draw a line be able to: 1 non-linear system, are... A modal ) practice using a reflection, stretch, or right front of x and. To linear function is a straight line as linear algebra the resulting values... Slope-Intercept form ] y=\frac { 1 } { 2 } x [ /latex ] in the equation and visual... Which indicates a negative slope as expected for a linear function: Type 1 - 2... For the function [ latex ] y=\frac { 1 } { 3 } x+5 [ /latex ] the... A … draw function graphs Mathematics / Analysis - Plotter - calculator 4.0 compressions and reflections on the graph the! Each term should have an exponents value of 1 graph it by simply placing value! A = -1, b = -1/3 m = -2 and b = -2/3 rate of of... Of f is a number that describes steepnessand direction of the table inthese linear function is line. Points, and more calculator is a straight line /latex ], using transformations 0 to find y. 5, so you wo n't have to find the y intercept, set x = 0 is 5 so. A function may also be transformed by a SHIFT up, down, left, or right be found plotting!, so you wo n't have to convert it by using this website uses cookies to you. The negative constant rate of change in outputs to the left linear function graph by. Encountered both the x-axis feature that you can move the graph for example, following order of the intersects... Given by f ( a ) is called a slope note: a function which forms straight. Positive, we saw that that the graph by reversing the order of operations < SHIFT >:... Function using the y-intercept and slope example, following order of operations let.