College algebra Linear Functions : Definition, Horizontal and Vertical Lines, Slope, Rate of Change, Slopeintercept Form, Point-slope Form, Parallel and Perpendicular Lines, Linear Regression (sections.3 and.4) Definition: Linear Equation: a linear equation in two variables is an equation that could be written in the form Ax + By = C where A, B, and C are real numbers and A and B are not both zero. Graphs of linear equations will be perfectly straight lines. Which of the following are linear equations? 3x 4y 5 y 4 Why would we say that A and B are not both zero? x 7 y 0 y 3x 4 6x 4x 5y You might try to write the linear equations above in the form Ax + By = C, which is called the standard form. We will investigate the slope-intercept and point-slope forms in this section. We will investigate graphs of linear equations here. The idea behind a graph is that it shows every single point that makes the equation true. Another way to say this is that the points satisfy the equation. Points are in the form (x, y).
Slope: The slope of a line tells you how slanted it is. Imagine walking up (or down) a line from left to right and you understand why that is important. Lines can go up from left to right. Some are steeper than others. Lines can go down from left to right. Some lines are parallel. Lines can be vertical or horizontal. Some lines are perpendicular. Imagine any two points on a line. Slope is the ratio of how far we go up (or down) to how far we go right (or left) to get from one point to the other point. As the steepness of the line changes, this ratio would change too. Formula for Slope: The slope between the two points (x, y ) and (x, y ) is m y x. y x Subscripts denote first and second points. slope rise run rise = difference of y values run = difference of x values
expl : Use the formula to find the slope of the line that goes through the points (, 4) and (3, ). Plot the points on the graph to check yourself. It does not matter which point you call (x, y ). Horizontal and Vertical Lines: Should the slope be negative or positive? Find the slope of these lines. Rather than using the formula from the previous page, use the quicker rise over run method. slope rise run Use what you found above to generalize about the slope of all vertical and horizontal lines. Slope of any vertical line = Slope of any horizontal line = 3
Slope-intercept Form of a Line: Any (non-vertical) line could be written in the form y = mx + b. Here, m is the slope and b is the y-intercept. It also helps to think of (x, y) as a generic point on the line. expl : Find the slope and y-intercept of the line y = -9.3x + 4.7. expl 3: Find the slope of the line x = 5. expl 4: Find the slope of the line y = -8. expl 5: Find the slope and y-intercept of the line x 3y =. Method : Find points on the line and use the y y formula m. x x Method : Rewrite the equation in y = mx + b form and pick out m. 4
Average Rate of change: The slope of a line is the difference of the y-values divided by the difference of the x-values. This ratio tells us how fast y is changing with respect to x, or average rate of change. expl 6: Find the slope of the line and write it as a average rate of change. Don t forget the units. Calculate the slope. y y m x x What are the units on top and bottom? What does this slope mean to the owner of this truck? Parallel and Perpendicular Lines: Optional Worksheet: Parallel and Perpendicular lines: This worksheet focuses on parallel and perpendicular lines and how their slopes are related. It also practices finding slope as rise over run. After doing the worksheet or from memory, complete the following information. Slopes of parallel lines are. Slopes of perpendicular lines are. The product of the slopes of perpendicular lines is. 5
expl 7: Find the slope of a line that is parallel to the line y = 5x + 3. expl 8: Find the slope of a line that is perpendicular to the line y = 5x + 3. Different Forms: A linear equation could be written in many different forms. Each form has its own advantages. We will use the various forms to write equations depending on what information we are given and our preferences. fairly easy to find intercepts General Equation Example Standard Form Ax + By = C 3x + 4y = Slope-Intercept y = mx + b y 3 x 3 Form 4 y y m x x 3 y 3 x 8 4 Point-Slope Form slope and y- intercept easy to pick out slope and one particular point (relatively) easy to pick out All of these equations describe the same line! By the way, point-slope form is sometimes hard to remember but it really is just our old friend, the formula for slope. Check out how we derive it below. slope formula y y m x x multiply by x x m x x y y y y y y m m x x x x flip it around obscure one of the point s subscripts 6 Here m is the slope and (x, x ) is a specific point on the line. You can think of (x, y) as a generic point on the line.
expl 9: Write the equation of the line with the given slope and y-intercept. m = 3, b = 3 Your equations should have x and y in place. Remember the line s equation tells us how the x and y values of every point are related. expl 0: Find the equation of the line that has a slope of 4 and passes through the point (, 3). Write your answer in slope-intercept form. You can use either the y y mx x or the y = mx + b form. expl : Write the equation of the line that passes through (8, 5) and has undefined slope. Which lines have undefined slope? 7
expl : Follow the steps below to find the equation of the line that passes through the points (6, ) and (8, 8). Write your answer in slope-intercept form. a.) Find m, the slope. You can use either the y y mx x or the y = mx + b form. Let s use y = mx + b this time. b.) Find b, the y-intercept. Use your slope and one of the points to find b. c.) Write your equation in the form y = mx + b with x and y in place. expl 3: Find the equation of the line that has a slope of 5 and goes through the point (6, 8). Write your final answer in slope-intercept form. You can use either the y y mx x or the y = mx + b form. expl 4: Margie has a pastry shop. For a certain kind of cupcake she produces, she has a variable cost of $.5 per cupcake and fixed costs of $00. Let x be the number of cupcakes she makes, and let C(x) be the total cost for these cupcakes. a.) Find an equation for C(x), her total cost. b.) How much will it cost Margie to produce 00 cupcakes? 8
Straight Line Graphs: So what makes the linear function graph as a straight line and other functions, like quadratic functions, not? Consider the functions below. Quadratic function Linear function x y = x 3 x y = 3x + -3 6-3 -7 - - -4 - - - - 0-3 0-5 8 3 6 3 Quickly graph it here using y = mx + b. What about the y-values in the linear function s table will force the graph to be a straight line? expl 5: Can you make up a table of values that would be graphed as a straight line? Complete the table below. x 0 y How do you assign y-values so that it graphs as a straight line? 3 4 5 9
Linear Regression: Take a look at the scatter plot below that shows the relationship between the time it takes to run the 400 meter dash and the year. Notice how the scatter plot takes on a linear pattern. If we were to find the equation of the line that best fit this pattern of points, we could use it to predict the time it takes to run the 400 meter dash in any given year. That is the idea of regression. Worksheet: Linear regression on your calculator: We will explore a couple of examples with step-by-step instructions on how to find the regression equations using the calculator. Coefficient of correlation, r: This number tells us how well the line fits the pattern of points and if the slope of the line is positive or negative. The coefficient of correlation ranges from - to. If r is negative, the line has a negative slope. If r is positive, the line has a positive slope. The closer r is to - or, the better the fit. Can you estimate r in the previous example? Is it positive or negative? What does that mean about the relationship between the time it takes to run 400 meters and the year? 0