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Exam 1: Linear Functions, Equations, and Inequalities

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Question 1

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Give the equations of both the horizontal and vertical lines passing through the point $(7,3)$ .

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Question 2

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Use the figure to solve each equation or inequality. (a) $f(x)=g(x)$ (b) $f(x)<g(x)$ (c) $f(x) \geq g(x)$ (d) $y_{2}-y_{1}=0$

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The table lists the average cost (in dollars) of tuition and fees at public four-year colleges for selected years. (Source: The College Board) (a) Find the least-squares regression line for the data. Give the correlation coefficient. (b) Use this line to predict the average annual cost of tuition and fees in 2007.

From 2004 to 2008 the federal debt (in billion of dollars) rose in approximately linear fashion. See the graph below. (Source: U.S. Office of Management and Budget) (a) Use the midpoint formula to approximate the debt during the year 2006. (b) Find the slope of the line and explain its meaning in the context of this situation.

Find the $x$ - and $y$ -intercepts of the line whose standard form is $3 x-y=5$ . What is the slope of this line?

For each of the functions, determine the (i) domain (ii) range (iii) $x$ -intercept(s) (iv) $y$ -intercept(s). (a) (b) (c)

Suppose that an empty circular wading pool has a radius of 7 feet. During a storm, rain falling at a rate of 1 inch per hour begins to fill the pool. A small drain at the bottom of the pool is capable of draining 35 gallons of water per hour. (a) Determine the number of cubic inches of water falling into the pool in one hour. (Hint: Each hour a layer of water 1 inch thick falls into the pool.) (b) One gallon of water equals about 231 cubic inches. Write a formula for a function $g$ that computes the gallons of water landing in the pool in $x$ hours. (c) How many gallons of water land in the pool during a 3 hour storm? (d) Will the drain be able to keep up with the rainfall? If not, how many such drains would be needed?

Give the equations of both the horizontal and vertical lines passing through the point $(3,0)$ .

Find the equation of the line passing through the point $(1,-4)$ and (a) parallel to the line with equation $y=2 x+3$ . (b) perpendicular to the line with equation $3 x+y=0$ .

Consider the linear function $f(x)=5(x+4)-(2 x+5)$ . (a) Solve the equation $f(x)=0$ analytically. (b) Solve the inequality $f(x) \leq 0$ analytically. (c) Graph $y=f(x)$ in an appropriate viewing window and explain how the graph supports your answers in parts (a) and (b).

Find the $x$ - and $y$ -intercepts of the line whose standard form is $3 x+7 y=-9$ . What is the slope of this line?

Use the screen to solve the equation or inequality. Here the function $y_{1}=f(x)$ is a linear function defined over the domain of real numbers. (a) $y_{1}=0$ (b) $y_{1}<0$ (c) $y_{1}>0$ (d) $y_{1} \leq 0$

Find the equation of the line passing through the point $(2,3)$ and (a) parallel to the line with equation $2 x+3 y=4$ . (b) perpendicular to the line with equation $y=\frac{1}{4} x+2$ .

Consider the linear function $f(x)=2(x-3)-\frac{1}{2}(6 x+2)$ . (a) Solve the equation $f(x)=0$ analytically. (b) Solve the inequality $f(x) \leq 0$ analytically. (c) Graph $y=f(x)$ in an appropriate viewing window and explain how the graph supports your answers in parts (a) and (b).

Consider the linear functions $f(x)=4-3 x+4(x+1)$ and $g(x)=3(x+1)-5(x+2)$ . (a) Solve $f(x)=g(x)$ analytically, showing all steps. Also, check analytically. (b) Graph $y_{1}=f(x)$ and $y_{2}=g(x)$ and use your result in part (a) to find the solution set of $f(x)<g(x)$ . Explain your answer. (c) Repeat part (b) for $f(x)>g(x)$ .

The table lists the Gross Domestic Product (GDP) from 2004 to 2008 in billions of dollars. (Source: U.S. Department of Commerce) (a) Find the least-squares regression line for the data. Give the correlation coefficient. (b) Use this line to predict the federal debt in 2009.

For the years 1996 to 2000, it was estimated that Medicare costs (in billions of dollars) would rise in a linear fashion. See the graph below. (Source: U.S. Office of Management and Budget) (a) Use the midpoint formula to approximate Medicare costs during the year 1998. (b) Find the slope of the line and explain its meaning in the context of this situation.

Find the equation of the line passing through the point $(2,-1)$ and (a) parallel to the line with equation $y=\frac{1}{2} x+3$ . (b) perpendicular to the line with equation $-3 x-y=4$ .

Give the equations of both the horizontal and vertical lines passing through the point $(4,-7)$ .

For each of the functions, determine the (i) domain (ii) range (iii) $x$ -intercept(s) (iv) $y$ -intercept(s). (a) (c)