Linear Functions

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Pre-Calculus › Linear Functions

Questions 1 - 10

What is the y-intercept of the line below?

Explanation

By definition, the y-intercept is the point on the line that crosses the y-axis. This can be found by substituting into the equation. When we do this with our equation,

$y = 3x + 4 = 3\cdot0 + 4 = 0 + 4 = 4$ .

Alternatively, you can remember form, a general form for a line in which is the slope and is the y-intercept.

What is the y-intercept of the line below?

Explanation

By definition, the y-intercept is the point on the line that crosses the y-axis. This can be found by substituting into the equation. When we do this with our equation,

$y = 3x + 4 = 3\cdot0 + 4 = 0 + 4 = 4$ .

Alternatively, you can remember form, a general form for a line in which is the slope and is the y-intercept.

What is the slope of the line below?

$y = \frac{2}{3}x + 6$

$\frac{2}{3}$

$-\frac{2}{3}$

$\frac{1}{6}$

Explanation

Recall slope-intercept form, or . In this form, is the slope and is the y-intercept. Given our equation above, the slope must be the coefficient of the x, which is $\frac{2}{3}$ .

What is the slope of the line below?

$y = \frac{2}{3}x + 6$

$\frac{2}{3}$

$-\frac{2}{3}$

$\frac{1}{6}$

Explanation

Recall slope-intercept form, or . In this form, is the slope and is the y-intercept. Given our equation above, the slope must be the coefficient of the x, which is $\frac{2}{3}$ .

What equation is perpendicular to $y = -\frac{2}{5}x+3$ and passes throgh ?

$y = \frac{5}{2}x - 1$

$y = \frac{2}{5}x - 2$

$y = 5x - \frac{1}{2}$

$y = \frac{5}{2}x - \frac{3}{2}$

$y = \frac{2}{5}x - \frac{4}{3}$

Explanation

First find the reciprocal of the slope of the given function.
$m_2 = -\frac{1}{m_1}$

$m_2 = \frac{5}{2}$

The perpendicular function is:

$y = \frac{5}{2}x+c$

Now we must find the constant, , by using the given point that the perpendicular crosses.

$4 = \frac{5}{2}(2)+c$

solve for :

$y = \frac{5}{2}x - 1$

What equation is perpendicular to $y = -\frac{2}{5}x+3$ and passes throgh ?

$y = \frac{5}{2}x - 1$

$y = \frac{2}{5}x - 2$

$y = 5x - \frac{1}{2}$

$y = \frac{5}{2}x - \frac{3}{2}$

$y = \frac{2}{5}x - \frac{4}{3}$

Explanation

First find the reciprocal of the slope of the given function.
$m_2 = -\frac{1}{m_1}$

$m_2 = \frac{5}{2}$

The perpendicular function is:

$y = \frac{5}{2}x+c$

Now we must find the constant, , by using the given point that the perpendicular crosses.

$4 = \frac{5}{2}(2)+c$

solve for :

$y = \frac{5}{2}x - 1$

Find the equation of the line with slope that passes through the point $\left(\frac{5}{4},3\right)$ .

Express your answer in form.

$y= 2x + \frac{1}{2}$

$y=\frac{5}{4}x +3$

None of the other answers.

$y = 3x +\frac{5}{4}$

$y = \frac{1}{2}x + 2$

Explanation

Since our slope is , we can plug it into right away giving .

To solve for , we plug our given point $\left(\frac{5}{4},3\right)$ in for and giving $3 = 2\frac{5}{4} +b$ .

This will simplify to $3-\frac{10}{4} = b$ , or $\frac{1}{2} = b$ after subtracting the fraction.

Hence our answer is $y = 2x + \frac{1}{2}$ after plugging our values for and in.

Find the equation of the line with slope that passes through the point $\left(\frac{5}{4},3\right)$ .

Express your answer in form.

$y= 2x + \frac{1}{2}$

$y=\frac{5}{4}x +3$

None of the other answers.

$y = 3x +\frac{5}{4}$

$y = \frac{1}{2}x + 2$

Explanation

Since our slope is , we can plug it into right away giving .

To solve for , we plug our given point $\left(\frac{5}{4},3\right)$ in for and giving $3 = 2\frac{5}{4} +b$ .

This will simplify to $3-\frac{10}{4} = b$ , or $\frac{1}{2} = b$ after subtracting the fraction.

Hence our answer is $y = 2x + \frac{1}{2}$ after plugging our values for and in.

Suppose Bob has 4 candies. He then earns candies at a rate of 14 candies per week. Which of the following formulas is the most reasonable rate to know how many candies Bob has on any given day?

$C=\frac{1}{2}d+4$

$C=\frac{2}{7}d+4$

$C=\frac{1}{7}d+4$

Explanation

Bob starts out with 4 candies. Write the equation.

Every week, he earns 14 candies. Every week has a total of 7 days. Divide the amount of candies by the number of days to determine the number of candies Bob earn in a day.

$\frac{14: \textup{candies}}{7: \textup{day}}=\frac{2: \textup{candies}}{1: \textup{day}}$

Bob then earns 2 candies per day. Let represent the number of candies per day. Finish the incomplete equation.

Suppose Bob has 4 candies. He then earns candies at a rate of 14 candies per week. Which of the following formulas is the most reasonable rate to know how many candies Bob has on any given day?

$C=\frac{1}{2}d+4$

$C=\frac{2}{7}d+4$

$C=\frac{1}{7}d+4$

Explanation

Bob starts out with 4 candies. Write the equation.

Every week, he earns 14 candies. Every week has a total of 7 days. Divide the amount of candies by the number of days to determine the number of candies Bob earn in a day.

$\frac{14: \textup{candies}}{7: \textup{day}}=\frac{2: \textup{candies}}{1: \textup{day}}$

Bob then earns 2 candies per day. Let represent the number of candies per day. Finish the incomplete equation.

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