Finding Intercepts Describe how to find the $x$-and $y$-intercepts of the graph of an equation.

Jacob C.

Numerade Educator

Verifying Points of Intersection How can you check that an ordered pair is a point of intersection of two graphs?

Jacob D.

Numerade Educator

match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).

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

Barbara P.

Numerade Educator

match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).

$y=\sqrt{9-x^{2}}$

James C.

Numerade Educator

match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).

$y=3-x^{2}$

Amy J.

Numerade Educator

match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).

$y=x^{3}-x$

James C.

Numerade Educator

Sketching a Graph by Point Plotting In Exercises $7-16,$ sketch the graph of the equation by point plotting.

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

A M.

Numerade Educator

Sketching a Graph by Point Plotting In Exercises $7-16,$ sketch the graph of the equation by point plotting.

$y=5-2 x$

Jacob C.

Numerade Educator

Sketching a Graph by Point Plotting In Exercises $7-16,$ sketch the graph of the equation by point plotting.

$y=4-x^{2}$

Linjun L.

Numerade Educator

Sketching a Graph by Point Plotting In Exercises $7-16,$ sketch the graph of the equation by point plotting.

$y=(x-3)^{2}$

Jacob C.

Numerade Educator

Sketching a Graph by Point Plotting In Exercises $7-16,$ sketch the graph of the equation by point plotting.

$y=|x+1|$

James C.

Numerade Educator

Sketching a Graph by Point Plotting In Exercises $7-16,$ sketch the graph of the equation by point plotting.

$y=|x|-1$

Jacob C.

Numerade Educator

Sketching a Graph by Point Plotting In Exercises $7-16,$ sketch the graph of the equation by point plotting.

$y=\sqrt{x}-6$

Robert S.

Numerade Educator

Sketching a Graph by Point Plotting In Exercises $7-16,$ sketch the graph of the equation by point plotting.

$y=\sqrt{x+2}$

Jacob C.

Numerade Educator

Sketching a Graph by Point Plotting In Exercises $7-16,$ sketch the graph of the equation by point plotting.

$y=\frac{3}{x}$

Steven G.

Numerade Educator

Sketching a Graph by Point Plotting In Exercises $7-16,$ sketch the graph of the equation by point plotting.

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

Jacob C.

Numerade Educator

Approximating Solution Points Using Technology In Exercises 17 and $18,$ use a graphing utility to graph the equation. Move the cursor along the curve to approximate the

unknown coordinate of each solution point accurate to two decimal places.

$$y=\sqrt{5-x}$$

(a) $(2, y)$

(b) $(x, 3)$

Jacob D.

Numerade Educator

Approximating Solution Points Using Technology In Exercises 17 and $18,$ use a graphing utility to graph the equation. Move the cursor along the curve to approximate the

unknown coordinate of each solution point accurate to two decimal places.

$$y=x^{5}-5 x$$

(a) $(-0.5, y)$

(b) $(x,-4)$

Jacob D.

Numerade Educator

Finding Intercepts In Exercises $19-28,$ find any intercepts.

$y=2 x-5$

Thomas H.

Numerade Educator

Finding Intercepts In Exercises $19-28,$ find any intercepts.

$y=4 x^{2}+3$

Jacob C.

Numerade Educator

Finding Intercepts In Exercises $19-28,$ find any intercepts.

$y=x^{2}+x-2$

Shaurya T.

Numerade Educator

Finding Intercepts In Exercises $19-28,$ find any intercepts.

$y^{2}=x^{3}-4 x$

Jacob C.

Numerade Educator

Finding Intercepts In Exercises $19-28,$ find any intercepts.

$y=x \sqrt{16-x^{2}}$

Jack R.

Numerade Educator

Finding Intercepts In Exercises $19-28,$ find any intercepts.

$y=(x-1) \sqrt{x^{2}+1}$

Jacob C.

Numerade Educator

Finding Intercepts In Exercises $19-28,$ find any intercepts.

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

Paul G.

Numerade Educator

Finding Intercepts In Exercises $19-28,$ find any intercepts.

$y=\frac{x^{2}+3 x}{(3 x+1)^{2}}$

Maral M.

Numerade Educator

Finding Intercepts In Exercises $19-28,$ find any intercepts.

$x^{2} y-x^{2}+4 y=0$

James C.

Numerade Educator

Finding Intercepts In Exercises $19-28,$ find any intercepts.

$y=2 x-\sqrt{x^{2}+1}$

Jacob C.

Numerade Educator

Testing for Symmetry In Exercises $29-40$ , test for symmetry with respect to each axis and to the origin.

$y=x^{2}-6$

James C.

Numerade Educator

Testing for Symmetry In Exercises $29-40$ , test for symmetry with respect to each axis and to the origin.

$y=9 x-x^{2}$

James C.

Numerade Educator

Testing for Symmetry In Exercises $29-40$ , test for symmetry with respect to each axis and to the origin.

$y^{2}=x^{3}-8 x$

Diego R.

Numerade Educator

Testing for Symmetry In Exercises $29-40$ , test for symmetry with respect to each axis and to the origin.

$y=x^{3}+x$

James C.

Numerade Educator

Testing for Symmetry In Exercises $29-40$ , test for symmetry with respect to each axis and to the origin.

$x y=4$

Amy J.

Numerade Educator

Testing for Symmetry In Exercises $29-40$ , test for symmetry with respect to each axis and to the origin.

$x y^{2}=-10$

James C.

Numerade Educator

Testing for Symmetry In Exercises $29-40$ , test for symmetry with respect to each axis and to the origin.

$y=4-\sqrt{x+3}$

James C.

Numerade Educator

Testing for Symmetry In Exercises $29-40$ , test for symmetry with respect to each axis and to the origin.

$x y-\sqrt{4-x^{2}}=0$

James C.

Numerade Educator

Testing for Symmetry In Exercises $29-40$ , test for symmetry with respect to each axis and to the origin.

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

James C.

Numerade Educator

Testing for Symmetry In Exercises $29-40$ , test for symmetry with respect to each axis and to the origin.

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

James C.

Numerade Educator

Testing for Symmetry In Exercises $29-40$ , test for symmetry with respect to each axis and to the origin.

$y=\left|x^{3}+x\right|$

Jacob D.

Numerade Educator

Testing for Symmetry In Exercises $29-40$ , test for symmetry with respect to each axis and to the origin.

$|y|-x=3$

Jacob D.

Numerade Educator

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.

$y=2-3 x$

Jacob D.

Numerade Educator

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.

$y=\frac{2}{3} x+1$

Jacob D.

Numerade Educator

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.

$y=9-x^{2}$

Paul-Yvann D.

Numerade Educator

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.

$y=2 x^{2}+x$

Jacob D.

Numerade Educator

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.

$y=x^{3}+2$

Jacob D.

Numerade Educator

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.

$y=x^{3}-4 x$

Jacob D.

Numerade Educator

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.

$y=x \sqrt{x+5}$

Timothy W.

Numerade Educator

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.

$y=\sqrt{25-x^{2}}$

Jacob D.

Numerade Educator

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.

$x=y^{3}$

Jacob D.

Numerade Educator

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.

$x=y^{4}-16$

Jacob D.

Numerade Educator

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.

$y=\frac{8}{x}$

Amy J.

Numerade Educator

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.

$y=\frac{10}{x^{2}+1}$

Jacob D.

Numerade Educator

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.

$y=6-|x|$

William C.

Numerade Educator

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.

$y=|6-x|$

Jacob D.

Numerade Educator

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.

$3 y^{2}-x=9$

Linxiao T.

Numerade Educator

Using Intercepts and Symmetry to Sketch a Graph In Exercises $41-56,$ find any intercepts and test for symmetry. Then sketch the graph of the equation.

$x^{2}+4 y^{2}=4$

Jacob D.

Numerade Educator

Finding Points of Intersection In Exercises $57-62,$ find the points of intersection of the graphs of the equations.

$x+y=8$

$4 x-y=7$

Georgiann A.

Numerade Educator

Finding Points of Intersection In Exercises $57-62,$ find the points of intersection of the graphs of the equations.

$3 x-2 y=-4$

$4 x+2 y=-10$

Jacob D.

Numerade Educator

Finding Points of Intersection In Exercises $57-62,$ find the points of intersection of the graphs of the equations.

$\begin{aligned} x^{2}+y &=15 \\-3 x+y &=11 \end{aligned}$

Catherine R.

Numerade Educator

$x=3-y^{2}$

$y=x-1$

Jacob D.

Numerade Educator

$x^{2}+y^{2}=5$

$x-y=1$

Amy J.

Numerade Educator

$x^{2}+y^{2}=16$

$x+2 y=4$

Jacob D.

Numerade Educator

Finding Points of Intersection Using Technology In Exercises $63-66$ , use a graphing utility to find the points of intersection of the graphs of the equations. Check your results

analytically.

$y=x^{3}-2 x^{2}+x-1$

$y=-x^{2}+3 x-1$

Amy J.

Numerade Educator

Finding Points of Intersection Using Technology In Exercises $63-66$ , use a graphing utility to find the points of intersection of the graphs of the equations. Check your results

analytically.

$y=x^{4}-2 x^{2}+1$

$y=1-x^{2}$

Jacob D.

Numerade Educator

Finding Points of Intersection Using Technology In Exercises $63-66$ , use a graphing utility to find the points of intersection of the graphs of the equations. Check your results

analytically.

$y=\sqrt{x+6}$

$y=\sqrt{-x^{2}-4 x}$

Amy J.

Numerade Educator

analytically.

$y=-|2 x-3|+6$

$y=6-x$

Jacob D.

Numerade Educator

Modeling Data The table shows the Gross Domestic Product, or GDP (in trillions of dollars), for 2009 through $2014 . \quad$ Source: U.S. Bureau of Economic Analysis)

$$\begin{array}{|c|c|c|c|c|c|c|}\hline \text { Year } & {2009} & {2010} & {2011} & {2012} & {2013} & {2014} \\ \hline \text { GDP } & {14.4} & {15.0} & {15.5} & {16.2} & {16.7} & {17.3} \\ \hline\end{array}$$

(a) Use the regression capabilities of a graphing utility to find a mathematical model of the form $y=a t+b$ for the data. In the model, $y$ represents the GDP (in trillions of dollars)

and $t$ represents the year, with $t=9$ corresponding to $2009 .$

(b) Use a graphing utility to plot the data and graph the model. Compare the data with the model.

(c) Use the model to predict the GDP in the year 2024 .

Amy J.

Numerade Educator

Modeling Data The table shows the numbers of cell phone subscribers (in millions) in the United States for selected years. (Source: CTIA-The Wireless Association)

$$\begin{array}{|c|c|c|c|c|}\hline \text { Year } & {2000} & {2002} & {2004} & {2006} \\ \hline \text { Number } & {109} & {141} & {182} & {233} \\ \hline \text { Year } & {2008} & {2010} & {2012} & {2014} \\ \hline \text { Number } & {270} & {296} & {326} & {355} \\ \hline\end{array}$$

(a) Use the regression capabilities of a graphing utility to find a mathematical model of the form $y=a t^{2}+b t+c$ for the data. In the model, $y$ represents the number of subscribers (in millions) and $t$ represents the year, with $t=0$ corresponding to $2000 .$

(b) Use a graphing utility to plot the data and graph the model. Compare the data with the model.

(c) Use the model to predict the number of cell phone subscribers in the United States in the year 2024 .

Jacob D.

Numerade Educator

Break-Even Point Find the sales necessary to break even $(R=C)$ when the cost $C$ of producing $x$ units is $C=2.04 x+5600$ and the revenue $R$ from selling $x$ units is $R=3.29 x .$

Jason K.

Numerade Educator

Using Solution Points For what values of $k$ does the graph of $y^{2}=4 k x$ pass through the point?

$$\begin{array}{ll}{\text { (a) }(1,1)} & {\text { (b) }(2,4)} \\ {\text { (c) }(0,0)} & {\text { (d) }(3,3)}\end{array}$$

Jacob D.

Numerade Educator

Using Intercepts Write an equation whose graph has intercepts at $x=-\frac{3}{2}, x=4,$ and $x=\frac{5}{2} .$ (There is more than one correct answer.)

Amy J.

Numerade Educator

Symmetry A graph is symmetric with respect to the $x$ -axis and to the $y$ -axis. Is the graph also symmetric with respect to the origin? Explain.

Jacob D.

Numerade Educator

Symmetry A graph is symmetric with respect to one axis and to the origin. Is the graph also symmetric with respect to the other axis? Explain.

Amy J.

Numerade Educator

Use the graphs of the two equations to answer the questions below.

(a) What are the intercepts for each equation?

(b) Determine the symmetry for each equation.

(c) Determine the point of intersection of the two equations.

Jacob D.

Numerade Educator

True or False? In Exercises $75-78$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $(-4,-5)$ is a point on a graph that is symmetric with respect to the $x$ -axis, then $(4,-5)$ is also a point on the graph.

Amy J.

Numerade Educator

True or False? In Exercises $75-78$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $(-4,-5)$ is a point on a graph that is symmetric with respect to the $y$ -axis, then $(4,-5)$ is also a point on the graph.

Jacob D.

Numerade Educator

True or False? In Exercises $75-78$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $b^{2}-4 a c>0$ and $a \neq 0,$ then the graph of

$$y=a x^{2}+b x+c$$

Amy J.

Numerade Educator

If $b^{2}-4 a c=0$ and $a \neq 0,$ then the graph of

$$y=a x^{2}+b x+c$$

has only one $x$ -intercept.

Jacob C.

Numerade Educator