0.2.1 Linear Functions

Linear Functions Easiest way to memorize the following is by their alphabetical order!! x y Input Output Domain Range

Linear Functions Can be expressed in many forms Equations y = 2x + 1 Graphically Table Format X Y 1 3 2 5 7 -1 Word Problem Gary charged $1 processing fee plus $2 per photo to develop pictures in his photography store

Review of Formulas Formula for Slope Standard Form Slope-intercept Form Point-Slope Form

Find the slope of a line through points (3, 4) and (-1, 6).

Change into standard form.

Change into slope-intercept form and identify the slope and y-intercept.

Write an equation for the line that passes through (-2, 5) and (1, 7): Find the slope: Use point-slope form:

x-intercepts and y-intercepts The intercept is the point(s) where the graph crosses the axis. To find an intercept, set the other variable equal to zero.

Horizontal Lines Slope is zero. Equation form is y = #. Write an equation of a line and graph it with zero slope and y-intercept of -2. y = -2 Write an equation of a line and graph it that passes through (2, 4) and (-3, 4). y = 4

Vertical Lines Slope is undefined. Equation form is x = #. Write an equation of a line and graph it with undefined slope and passes through (1, 0). x = 1 Write an equation of a line that passes through (3, 5) and (3, -2). x = 3

Graphing Lines graph a line. Using x and y intercepts: *You need at least 2 points to graph a line. Using x and y intercepts: Find the x and y intercepts Plot the points Draw your line

Graph using x and y intercepts 2x – 3y = -12 x-intercept 2x = -12 x = -6 (-6, 0) y-intercept -3y = -12 y = 4 (0, 4)

Graph using x and y intercepts 6x + 9y = 18 x-intercept 6x = 18 x = 3 (3, 0) y-intercept 9y = 18 y = 2 (0, 2)

Graphing Lines Using slope-intercept form y = mx + b: Change the equation to y = mx + b. Plot the y-intercept. Use the numerator of the slope to count the corresponding number of spaces up/down. Use the denominator of the slope to count the corresponding number of spaces left/right. Draw your line.

Graph using slope-intercept form y = -4x + 1: y-intercept (0, 1) Slope m = -4 = -4 1

Graph using slope-intercept form 3x - 4y = 8 y = 3x - 2 4 y-intercept (0, -2) Slope m = 3 4

Parallel Lines **Parallel lines have the same slopes. Find the slope of the original line. Use that slope to graph your new line and to write the equation of your new line.

Graph a line parallel to the given line and through point (0, -1): Slope = 3 5

Write the equation of a line parallel to 2x – 4y = 8 and containing (-1, 4): – 4y = - 2x + 8 y = 1x - 2 2 Slope = 1 y - 4 = 1(x + 1) 2

Perpendicular Lines **Perpendicular lines have the opposite reciprocal slopes. Find the slope of the original line. Change the sign and invert the numerator and denominator of the slope. Use that slope to graph your new line and to write the equation of your new line.

Graph a line perpendicular to the given line and through point (1, 0): Slope =-3 4 Perpendicular Slope= 4 3

Write the equation of a line perpendicular to y = -2x + 3 and containing (3, 7): Original Slope= -2 Perpendicular Slope = 1 2 y - 7 = 1(x - 3) 2

Write the equation of a line perpendicular to 3x – 4y = 8 and containing (-1, 4): -4y = -3x + 8 y - 4 = -4(x + 1) 3 Slope= 3 4 Perpendicular Slope = -4 3