Reteach 2-3. Graphing Linear Functions. 22 Holt Algebra 2. Name Date Class

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1 -3 Graphing Linear Functions Use intercepts to sketch the graph of the function 3x 6y 1. The x-intercept is where the graph crosses the x-axis. To find the x-intercept, set y 0 and solve for x. 3x 6y 1 3x 6(0) 1 3x 1 x 4 The y-intercept is where the graph crosses the y-axis. To find the y-intercept, set x 0 and solve for y. 3x 6y 1 3(0) 6y 1 6y 1 y The y-intercept occurs at the point 0,. Plot the points ( 4, 0) and ( 0, ). Draw a line connecting the points. The x-intercept occurs at the point 4, 0. Find the intercepts and graph each line. 1. 3x y 6 a. 3x 6 x-intercept b. 3 y 6 y-intercept. 6x 3y 1 a. 6x 3 1 x-intercept b. 6 3y 1 y-intercept Holt Algebra

2 -3 Graphing Linear Functions (continued) Use the slope and the y-intercept to graph a linear function. To write y x 6 in slope-intercept form, solve for y. y x 6 x x y x 6 y x 6 y x 3 Compare y x 3 to y mx b. m, so the slope is. b 3, so the y-intercept is 3. y mx b is the slope-intercept form. m represents the slope and b represents the y-intercept. Write each function in slope-intercept form. Use m and b to graph. 3. x y 1 a. y x 4. y x 1 a. y b. m b. m c. b c. b 3 Holt Algebra

3 -4 Writing Linear Functions Write the equation of the line shown in the graph in slope-intercept form. Slope-intercept form: y mx b The point, 4 lies on the line. From 0, 1, move 3 units down, or a rise of 3 units, and units right, or a run of units, to, 4. m rise run 3 3 Note that when the rise is a drop the slope is negative. Substitute m 3 and b 1 into y mx b to get the equation y 3 x 1. Write the equation of each line in slope-intercept form b b b m rise run 1 m rise run 1 m rise run y x y y 30 Holt Algebra

4 -4 Writing Linear Functions (continued) The slopes of parallel and perpendicular lines have a special relationship. The slopes of parallel lines are equal. y x 1 and y x are parallel lines since both equations have a slope of. Note: The slopes of parallel vertical lines are undefined. The slopes of perpendicular lines are negative reciprocals. Their product is 1. y x 1 and y x 1 are perpendicular since 1. The point-slope form of the equation of a line is y y 1 m x x 1. The line has slope m and passes through the point x 1, y 1. Write the equation of the line perpendicular to y x through, 5. 3 Substitute values for m and x 1, y 1 in y y 1 m x x 1. x 1, y 1, 5, so x 1, y 1 5, and m 3 y y 1 m x x 1 y 5 3 x y 5 3x 6 y 3x 11 The negative reciprocal of 3 is 3 because Write the equation of each line. 4. parallel to y 4x 3 through the point 1, m x 1, y 1, y y 1 m x x 1 y x y 5. perpendicular to y x 4 through the point 1, 1 m x 1, y 1, y y 1 m x x 1 y x y 31 Holt Algebra

5 -6 Transforming Linear Functions Translating linear functions vertically or horizontally changes the intercepts of the function. It does NOT change the slope. Let f x 3x 1. Read the rule for each translation. Horizontal Translation B Think: Add to x, go west. Use f x r f x h. Translation units right c g x f x g x 3 x 1 Rule: g x 3x 5 Translation units left V h x f x f x h x 3 x 1 Rule: h x 3x 7 Let f x x 1. Write the rule for g x. Vertical Translation n Think: Add to y, go high. Use f x r f x k. Translation units up M g x f x g x 3x 1 Rule: g x 3x 3 Translation units down m h x f x f x h x 3x 1 Rule: h x 3x 1 1. horizontal translation 5 units right. vertical translation 4 units down g x f x g x f x g x x 1 g x g x g x 3. vertical translation 3 units up 4. horizontal translation 1 unit left g x f x g x f x g x g x 1 g x g x 5. vertical translation 7 units down 6. horizontal translation 9 units right 7. vertical translation 1 unit up 8. horizontal translation unit to the left 46 Holt Algebra

6 -6 Transforming Linear Functions (continued) Compressing or stretching linear functions changes the slope. Let f x 3x 1. Read the rule for each translation. Horizontal stretch or compression by a factor of b Use f x r f b x. Horizontal stretch by a factor of g x f b x f x g x 3 x 1 Rule: g x 3 x 1 Horizontal compression by a factor of h x f b x f 1 h x 3 x 1 Rule: h x 6x 1 x f x Vertical stretch or compression by a factor of a Use f x r a f x. Vertical stretch by a factor of g x a f x f x g x 3x 1 Rule: g x 6x Vertical compression by a factor of h x a f x f x h x 3x 1 Rule: h x 3 x Let f x x 1. Write the rule for g x. 7. vertical compression by a factor of 4 f x r g x g x 8. horizontal stretch by a factor of 3 4 f x f x r f 1 3 x 4 x 1 g x 1 3 x 1 x 1 4 g x 3 x 1 9. horizontal compression by a factor of 3 f x r 3 1 x vertical stretch by a factor of 5 f x r 5 x 1 g x 6x 1 g x r g x 10x 5 47 Holt Algebra

7 -7 Curve Fitting with Linear Models Use a scatter plot to identify a correlation. If the variables appear correlated, then find a line of fit. Positive correlation Negative correlation No correlation The table shows the relationship between two variables. Identify the correlation, sketch a line of fit, and find its equation. x y Step 1 Step Step 3 Step 4 Step 5 Make a scatter plot of the data. As x increases, y decreases. The data is negatively correlated. Use a straightedge to draw a line. There will be some points above and below the line. Choose two points on the line to find the equation: 1, 16 and 7,. Use the points to find the slope: m change in y change in x Use the point-slope form to find the equation of a line that models the data. y y 1 m x x 1 y 7 3 x 7 y 7 x 18 3 Use the scatter plot of the data to solve. 1. The correlation is Positive.. Choose two points on the line and find the slope. 4, 8 and 6, 11 ; m 3 3. Find the equation of a line that models the data. y 3 x 54 Holt Algebra

8 -7 Curve Fitting with Linear Models (continued) A line of best fit can be used to predict data. Use the correlation coefficient, r, to measure how well the data fits. If r is near 1, data is modeled by a line with a negative slope. 1 r 1 If r is near 0, data has no correlation. If r is near 1, data is modeled by a line with a positive slope. Use a graphing calculator to find the correlation coefficient of the data and the line of best fit. Use STAT EDIT to enter the data. x y Use LinReg from the STAT CALC menu to find the line of best fit and the correlation coefficient. LinReg y ax b a.0 b r.9308 r.9648 The correlation coefficient is The data is very close to linear with a negative slope. Use the linear regression model to predict y when x 3.5. y.x y y Use a calculator and the scatter plot of the data to solve. 4. Find the correlation coefficient, r Find the equation of the line of best fit. y 1.38x.9 6. Predict y when x.6. y Predict y when x 5.3. y Holt Algebra