2.2 Graphs of Linear Functions

Find a point on the graph we drew in the example that has a negative x-value.

Find the x-intercept of $f \left(x\right) = \frac{1}{2} x - 3.$

Find the x-intercept of $f \left(x\right) = \frac{1}{4} x - 4.$

Given the functions below, identify the functions whose graphs are a pair of parallel lines and a pair of perpendicular lines.

Find a line parallel to the graph of $f \left(x\right) = 3 x + 6$ that passes through the point $\left(3 , \textrm{ }\text{0}\right) .$

Find the equation of a line perpendicular to $f \left(x\right) = 3 x + 3$ that passes through the point $\left(3 , \textrm{ }\text{0}\right) .$

Given the function $h \left(x\right) = 2 x - 4 ,$ write an equation for the line passing through $\left(0 , 0\right)$ that is

1. ⓐ parallel to $h \left(x\right)$
2. ⓑ perpendicular to $h \left(x\right)$

A line passes through the points $\left(- 2 , \textrm{ }\text{6}\right)$ and $\left(4 , 5\right) .$ Find the equation of a perpendicular line that passes through the point $\left(4 , 5\right) .$

A line passes through the points, $\left(−2 , −\text{15}\right)$ and $\left(2, −3\right) .$ Find the equation of a perpendicular line that passes through the point, $\left(6 , 4\right) .$

Find the point of intersection of the lines $h \left(t\right) = 3 t - 4$ and $j \left(t\right) = 5 - t .$

Look at the graph in the figure and identify the following for the function $j \left(t\right) :$

1. ⓐ y-intercept
2. ⓑ x-intercept(s)
3. ⓒ slope
4. ⓓ Is $j \left(t\right)$ parallel or perpendicular to $h \left(t\right)$ (or neither)?
5. ⓔ Is $j \left(t\right)$ an increasing or decreasing function (or neither)?
6. ⓕ Write a transformation description for $j \left(t\right)$ from the identity toolkit function $f \left(x\right) = x .$

A company sells sports helmets. The company incurs a one-time fixed cost for $250,000. Each helmet costs $120 to produce, and sells for $140.

1. ⓐ Find the cost function, $C ,$ to produce $x$ helmets, in dollars.
2. ⓑ Find the revenue function, $R ,$ from the sales of $x$ helmets, in dollars.
3. ⓒ Find the break-even point, the point of intersection of the two graphs $C$ and $R .$

If the graphs of two linear functions are parallel, describe the relationship between the slopes and the y-intercepts.

If a horizontal line has the equation $f \left(x\right) = a$ and a vertical line has the equation $x = a ,$ what is the point of intersection? Explain why what you found is the point of intersection.

Explain how to find a line perpendicular to a linear function that passes through a given point.

Determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:

$\begin{matrix}3 y + x = 12 \\ - y = 8 x + 1\end{matrix}$

Determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:

$\begin{matrix}6 x - 9 y = 10 \\ 3 x + 2 y = 1\end{matrix}$

Determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:

$\begin{matrix}y = \frac{3}{4} x + 1 \\ - 3 x + 4 y = 1\end{matrix}$

Find the x- and y-intercepts of each equation

$g \left(x\right) = 2 x + 4$

Find the x- and y-intercepts of each equation

$k \left(x\right) = - 5 x + 1$

Find the x- and y-intercepts of each equation

$7 x + 2 y = 56$

Use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?

1. Line 1: Passes through $\left(−8 , −55\right)$ and $\left(10 , 89\right)$
2. Line 2: Passes through $\left(9 , −44\right)$ and $\left(4 , −14\right)$

Use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?

1. Line 1: Passes through $\left(1 , 7\right)$ and $\left(5 , 5\right)$
2. Line 2: Passes through $\left(−1 , −3\right)$ and $\left(1 , 1\right)$

Use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?

1. Line 1: Passes through $\left(2 , 5\right)$ and $\left(5 , −1\right)$
2. Line 2: Passes through $\left(−3 , 7\right)$ and $\left(3 , −5\right)$

Use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?

Write an equation for a line parallel to $g \left(x\right) = 3 x - 1$ and passing through the point $\left(4 , 9\right) .$

Use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?

Write an equation for a line perpendicular to $p \left(t\right) = 3 t + 4$ and passing through the point $\left(3 , 1\right) .$

Use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?

Find the point at which the line $f \left(x\right) = 2 x + 5$ intersects the line $g \left(x\right) = - 3 x - 5.$

Use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?

Use algebra to find the point at which the line $f \left(x\right) = \frac{7}{4} x + \frac{457}{60}$ intersects the line $g \left(x\right) = \frac{4}{3} x + \frac{31}{5} .$

$f \left(x\right) = - 3 x - 1$

$f \left(x\right) = 2$

$f \left(x\right) = 3 x + 2$

Sketch the graph of each equation.

If $g \left(x\right)$ is the transformation of $f \left(x\right) = x$ after a vertical compression by $\frac{3}{4} ,$ a shift right by 2, and a shift down by 4

1. ⓐ Write an equation for $g \left(x\right) .$
2. ⓑ What is the slope of this line?
3. ⓒ Find the y-intercept of this line.

Find the point of intersection of each pair of lines if it exists. If it does not exist, indicate that there is no point of intersection.

$\begin{matrix}y = \frac{3}{4} x + 1 \\ - 3 x + 4 y = 12\end{matrix}$

Find the point of intersection of each pair of lines if it exists. If it does not exist, indicate that there is no point of intersection.

$\begin{matrix}2 x = y - 3 \\ y + 4 x = 15\end{matrix}$

Find the point of intersection of each pair of lines if it exists. If it does not exist, indicate that there is no point of intersection.

$\begin{matrix}5 x + 3 y = - 65 \\ x - y = - 5\end{matrix}$

Find the equation of the line perpendicular to the line $g \left(x\right) = - 0. \text{01} x +\text{2} .\text{01}$ through the point $\left(\text{1} , \textrm{ }\text{2}\right) .$

Use the functions $f \left(x\right) = - 0. \text{1} x +\text{200}\textrm{ }\text{and}\textrm{ } g \left(x\right) = 20 x + 0.1.$

Where is $f \left(x\right)$ greater than $g \left(x\right) ?$ Where is $g \left(x\right)$ greater than $f \left(x\right) ?$

A cell phone company offers two plans for minutes.

1. Plan A: $20 per month and $1 for every one hundred texts.
2. Plan B: $50 per month with free unlimited texts.

How many texts would you need to send per month for plan B to save you money?