Graphs

In Problems 69–76, (a) find the intercepts of each equation, (b) test each equation for symmetry with respect to the x-axis, the y-axis, and the origin, and (c) graph each equation by hand by plotting points. Be sure to label the intercepts on the graph and use any symmetry to assist in drawing the graph. Verify your results using a graphing utility.

$y=x^3-4x$

In Problems 69–76, (a) find the intercepts of each equation, (b) test each equation for symmetry with respect to the x-axis, the y-axis, and the origin, and (c) graph each equation by hand by plotting points. Be sure to label the intercepts on the graph and use any symmetry to assist in drawing the graph. Verify your results using a graphing utility.

$y=x^3-x$

Given that the point $(1,2)$ is on the graph of an equation that is symmetric with respect to the origin, what other point is on the graph?

If the graph of an equation is symmetric with respect to the $x$-axis and $2$ is a $y$-intercept, name another $y$-intercept.

Microphones In studiosand on stages, cardioid microphones are often preferred for the richness they add to voices and for their ability to reduce the level of sound from the sides and rear of the microphone. Suppose one such cardioid pattern is given by the equation ${\left(x^2+y^2-x\right)}^2=x^2+y^2$
.

(a) Find the intercepts of the graph of the equation.

(b) Test for symmetry with respect to the $x$-axis, $y$-axis, and origin.

Solar Energy The solar electric generating systems at Kramer Junction, California, use parabolic troughs to heat a heat-transfer fluid to a high temperature. This fluid is used to generate steam that drives a power conversion system to produce electricity. For troughs $7.5$ feet wide, an equation for the cross-section is $16y^2=120x-225$.

(a) Find the intercepts of the graph of the equation.

(b) Test for symmetry with respect to the $x$-axis, $y$-axis, and origin.

Draw a graph of an equation that contains two $x$-intercepts; at one the graph crosses the $x$-axis, and at the other the graph touches the $x$-axis.

Make up an equation with the intercepts $(2,0),(4,0)$, and $(0,1)$. Compare your equation with a friend's equation. Comment on any similarities.

Draw a graph that contains the points $(0,1),(1,3)$ , and $(3,5)$ and is symmetric with respect to the $x$-axis. Compare your graph with those of other students. Are most of the graphs almost straight lines? How many are "curved"? Discuss the various ways that these points might be connected.

An equation is being tested for symmetry with respect to the $x$-axis, the $y$-axis, and the origin. Explain why, if two of these symmetries are present, the remaining one must also be present.

Draw a graph that contains the points $(-2,5),(-1,3)$, and $(0,2)$ that is symmetric with respect to the $y$-axis. Compare your graph with those of other students; comment on any similarities. Can a graph contain these points and be symmetric with respect to the $x$-axis? The origin? Why or why not?

Open the y-axis symmetry applet. Move point A around the Cartesian plane with your mouse. How are the coordinates of point A and the coordinates of point B related?

Open the x-axis symmetry applet. Move point A around the Cartesian plane with your mouse. How are the coordinates of point A and the coordinates of point B related?

Open the origin symmetry applet. Move point A around the Cartesian plane with your mouse. How are the coordinates of point A and the coordinates of point B related?

 Solve the equation $2x^2 + 5x + 2 = 0.$

Solve the equation $2x+3 = 4(x - 1)+1$

To solve an equation of the form {expression in x = 0}

using a graphing utility, we graph $Y_1$ = {expression in x}

and use ________ to determine each x-intercept of the

graph.

True or False

 In using a graphing utility to solve an

equation, exact solutions are always obtained.

$x^3 - 4x + 2 = 0$

$x^3 - 8x + 1 = 0$

In Problems 5–16, use a graphing utility to approximate the real solutions, if any, of each equation rounded to two decimal places. All solutions lie between - 10 and 1$-2x^4+ 5 = 3x - 2$

In Problems 5–16, use a graphing utility to approximate the real solutions, if any, of each equation rounded to two decimal places. All solutions lie between - 10 and 10. $-x^4 + 1 = 2x^2 - 3$

In Problems 5–16, use a graphing utility to approximate the real solutions, if any, of each equation rounded to two decimal places. All solutions lie between - 10 and 10.$x^4- 2x^3+ 3x - 1 = 0$

.In Problems 5–16, use a graphing utility to approximate the real solutions, if any, of each equation rounded to two decimal places. All solutions lie between - 10 and 10.$3x^4- x^3+ 4x^2 - 5 = 0$

In Problems 5–16, use a graphing utility to approximate the real solutions, if any, of each equation rounded to two decimal places. All solutions lie between - 10 and 10.$-x^3- \frac{5}{3}{x}^2+ \frac{7}{2} x + 2 = 0$

In Problems 5–16, use a graphing utility to approximate the real solutions, if any, of each equation rounded to two decimal places. All solutions lie between - 10 and 10.$-x^4 + 3x^3+ \frac{7}{3} x^2 - \frac{15}{2} x + 2 = 0$

In Problems 5–16, use a graphing utility to approximate the real solutions, if any, of each equation rounded to two decimal places. All solutions lie between - 10 and 10.$- \frac{2}{3} x^4 - 2x^3 + \frac{5}{2} x = -\frac{2}{3} x^2+ 1 2$

In Problems 5–16, use a graphing utility to approximate the real solutions, if any, of each equation rounded to two decimal places. All solutions lie between - 10 and 10.$\frac{1}{4}{x}^3 - 5x = \frac{1}{5}{x}^2- 4$

In Problems 5–16, use a graphing utility to approximate the real solutions, if any, of each equation rounded to two decimal places. All solutions lie between - 10 and 10.$-3x^4 + 8x^2 - 2x - 9 = 0$

In Problems 17–36, solve each equation algebraically. Verify your solution using a graphing utility.$2(3 + 2x) = 3(x - 4)$

In Problems 17–36, solve each equation algebraically. Verify your solution using a graphing utility.$3(2 - x) = 2x -1$

In Problems 17–36, solve each equation algebraically. Verify your solution using a graphing utility.$8x-\left(2x+1\right)=3x-13$

In Problems 17–36, solve each equation algebraically. Verify your solution using a graphing utility..$5-\left(2x-1\right)=10-x$

In Problem 17-36, solve each equation algebraically. Verify your solution using a graphing utility.

$\frac{x+1}{3\hspace{0.17em}}+\frac{x+2}{7}=5$

In Problem 17-36, solve each equation algebraically. Verify your solution using a graphing utility.

$\frac{2x+1}{3}+16=3x$

In Problem 17-36, solve each equation algebraically. Verify your solution using a graphing utility.

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

In Problem 17-36, solve each equation algebraically. Verify your solution using a graphing utility.

$\frac{4}{y}-5=\frac{18}{2y}$

In Problem 17-36, solve each equation algebraically. Verify your solution using a graphing utility.

$\left(x+7\right)\left(x-1\right)={\left(x+1\right)}^2$

In Problem 17-36, solve each equation algebraically. Verify your solution using a graphing utility.

$\left(x+2\right)\left(x-3\right)={\left(x-3\right)}^2$

In Problem 17-36, solve each equation algebraically. Verify your solution using a graphing utility.

$x^2-3x-28=0$

In Problem 17-36, solve each equation algebraically. Verify your solution using a graphing utility.

$x^2-7x-18=0$

In Problem 17-36, solve each equation algebraically. Verify your solution using a graphing utility.

$3x^2=4x+4$

In Problem 17-36, solve each equation algebraically. Verify your solution using a graphing utility.

$5x^2=13x+6$

In Problem 17-36, solve each equation algebraically. Verify your solution using a graphing utility.

$x^3+x^2-4x-4=0$

In Problem 17-36, solve each equation algebraically. Verify your solution using a graphing utility.
$x^3+2x^2-9x-18=0$

In Problem 17-36, solve each equation algebraically. Verify your solution using a graphing utility. 

$\sqrt{x+1}=4$

In Problem 17-36, solve each equation algebraically. Verify your solution using a graphing utility. 

$\sqrt{x-2}=3$

In Problem 17-36, solve each equation algebraically. Verify your solution using a graphing utility. 

$\frac{2}{x+2}+\frac{3}{x-1}=-\frac{8}{5}$

In Problem 17-36, solve each equation algebraically. Verify your solution using a graphing utility. 

$\frac{1}{x+1}-\frac{5}{x-4}=\frac{21}{4}$

The slope of a vertical line is ________ ; the slope of a horizontal line is _______.