Linear Functions Vincent J Motto University of Hartford

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. For example

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 * 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: Slope m = -4 = -4 1 y-intercept (0, 1)

Graph using slope-intercept form 3x - 4y = 8 Slope m = 3 4 y-intercept (0, -2) y = 3x - 2 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 Slope =1 2 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 y - 7 = 1(x - 3) 2 Perpendicular Slope =1 2

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

Using Your Calculator The TI-89 Calculator is preferred. Follow the link below to acquire the skill of graphing. Graphing with the TI-89 Graphing with the Ti-83/84

Slope Rate of Change

Average Change Defined equations in terms of their changes – Linear Equations -> constant change – Exponential  constant percentage change Will use this concept motivate derivatives

Example 4 The problem: Enrique earns $6.00 per hour working at Quikee Mart. He is saving his wages to buy a 3GB iPOD. The iPOD is on sale for $ How many hours must Enrique work so he will have enough money to buy his iPOD? Step 1 - Make a table Step 2 Figure how much the domain and range values are changing. For the domain, you may use multiples of 5 to help you find the number of hours he needs to work How do I know that this rate of change is constant? The ANSWER – Enrique must work 35 hours to earn his iPOD. Make a table to help find the solution Hours worked Amount earned $30 $60 $90 $120 $150 $180 $210

Average Change Rate of change – Difference between two values Percentage change – Difference between two values as a percentage of the original value Average change – Change per unit of time

Average Change Let the table below describe the function r(t) = pool sales MonthSales

Average Change

Rate of change Percent change Average change (slope)

Average Change Rate of change Rate of change from April to August

Average Change r(8) - r(4) Rate of change from April to August

Average Change r(8) - r(4) Average rate of change from April to August

Average Change Average rate of change from April to August r(8) - r(4) 8-4

Average Change Average rate of change from April to August r(8) - r(4) 8-4