# 2.3 – Linear Functions

## Objectives

• Describe linear functions with words, graphically, with a table of values, or with an algebraic expression.
• Identify and interpret the slope and intercept of a line.
• Write equations of lines.

## Key Terms & Notes

Axes & Intercepts
• x-axis: The horizontal axis in a Cartesian coordinate system.
• y-axis: The vertical axis in a Cartesian coordinate system.
• x-intercept: A point where the graph of a function crosses the x-axis.
• This is the graph coordinate (x, 0).
• y-intercept: A point where the graph of a function crosses the y-axis.
• This is the graph coordinate (0, y).
• A function has at most one y-intercept (remember the vertical line test).
• The y-intercept of the line with equation y = mx + b is the point (0, b).
Linear Functions
• A linear function is a function whose graph is a line.
• A line is written in three forms:
• Standard form: ax + by = c
• Slope-Intercept form: y = mx + b
• Point-Slope form: $y-y_1=m(x-x_1)$

Slope
• Slope is the measure of the steepness of a line.
• Slope equals rise divided by run for any two points on the line.
• Rise over run, written as $\frac{rise}{run}$
• Change ($\Delta$) in y over change ($\Delta$) x, written as $\frac{\Delta y}{\Delta x}$

• Using slope to write point-slope
• Once you find the slope, put it into the point-slope form equation.
• You can use ANY point on the line to write the formula for this line in point-slope form.

• Rise and Fall
• A line that rises from left to right has a positive slope.
• A line that falls from left to right has a negative slope.
• A line that is horizontal has a zero slope.
• A line that is vertical has an undefined slope.
• You can’t divide a number (the rise) by zero (the run).
Parallel & Perpendicular Lines
• Parallel lines have equal slopes.
• Ex. The following lines are parallel because they have the same slope.
• $y=3x+4$
• $y=3x-2$
• Perpendicular lines have negative reciprocal slopes.
• Ex. The following lines are perpendicular because their slopes are negative reciprocals of one another.  This means that the slopes are flipped and one is positive while the other is negative.
• $y=3x+4$
• $y=- \frac{1}{3}-2$

## Examples

 Ex 1. What is the slope of a line is parallel to $y=\frac{1}{2}x+3$? Answer: $\frac{1}{2}$ Ex 2. What is the graph of the line $y=-3x-2$? Answer: There is a negative slope and a negative y-intercept, so the line crosses the y-axis at -2 and slopes down from left to right. Ex 3. What is the slope of the function, represented by the table of values below? Answer: Pick any two points in the chart and use the slope formula: $\frac{y_2-y_1}{x_2-x_1}$ $\frac{6-0}{0-3}=\frac{6}{-3}=-2$ Ex 4. Which of the following is the equation of a line that passes through the point (1,4) and is parallel to the x-axis? Answer: The x-axis has a zero slope, so substitute into the point-slope form: $y-4=0(x-1)$. This becomes $y-4=0$, which simplifies to $y=4$. Ex 5. Which of the following is the equation of a line that passes through the points (0,6) and (2,10)? To solve this, do the following steps: Find the slope: $m=\frac{10-6}{2-0}$, which simplifies to $\frac{4}{2}=2$. Put the slope and ONE of the points into point-slope form: $y-6=2(x-0)$. Use distribution and inverse operations to convert to slope-intercept form: $y-6=2x$, and then again to $y=2x+6$. Answer: $y=2x+6$ Ex 6. Jenny is selling books during the summer to earn money for college. She earns a commission on each sale but has to pay for her own expenses. After a month of driving from neighborhood to neighborhood and walking door-to-door, she figures out that her weekly earnings are approximately a linear function of the number of doors she knocks on. She writes the equation of the function like this: $E(x)=20x-50$, where x is the number of doors she knocks on during the week and E(x) is her earnings for the week in dollars. What does the slope of Jenny’s function represent? Answer: for each additional door she knocks on, her earnings will increase by $20. This is because the slope represents how much money she makes PER (each) door she knocks on. For every door, she makes$20 more dollars.  You can multiply 20 times the number of doors knocked on.  That’s why it is written as 20 times x. The $50 represents her base salary (and therefore, her expenses) for the week… it is not the slope. If she doesn’t knock on any doors, she will actually lose (minus) the$50 and not be able to pay for her expenses. Ex 7. The ordered pairs below represent a relation between x and y. (-3,0), (-2,4), (-1,8), (0,12), (1,16), (2,20) Could this set of ordered pairs have been generated by a linear function? Answer: Yes, because the relative difference between y-values and x-values is the same no matter which pairs of (x, y) values you use to calculate it. In other words, the slope is the same for any two points you calculate.  So, you have a line, which is represented by a linear function!