Chapter 1, Problem 16T

(a)

To determine

To sketch: The graph of equation $y=x^2-4$.

Explanation of Solution

The given equation is $y=x^2-4$.

Substitute different values of x to get corresponding values of y in above equation and make a table,

x	$y=x^2-4$	$\left(x,y\right)$
$\pm 2$	0	$\left(2,0\right)$, $\left(-2,0\right)$
$\pm 1$	$-3$	$\left(1,-3\right)$, $\left(-1,-3\right)$
0	$-4$	$\left(0,-4\right)$

Plot points from above table and connect them by a smooth curve which is shown in Figure (1),

Figure (1)

Figure (1) shows the graph of equation $y=x^2-4$ that is a parabola.

(b)

To determine

To find: The x-intercepts and y-intercepts of graph of the equation $y=x^2-4$.

Answer to Problem 16T

The values of x-intercepts are $-2$, 2 and value of y-intercept is $-4$.

Explanation of Solution

Given:

The equation is,

$y=x^2-4$. (1)

Calculation:

Substitute 0 for y in equation (1) and solve for x to get the value of x-intercept,

$\begin{array}{c}0=x^2-4\\ x^2=4\\ x=\pm \sqrt{4}\\ x=\pm 2\end{array}$

The x-intercept is $-2$ and 2.

Substitute 0 for x in given equation and solve for y to get the value of y-intercept,

$\begin{array}{l}y={\left(0\right)}^2-4\\ y=-4\end{array}$

The y-intercept is $-4$.

The graph intersects the x-axis at $x=-2$, $x=2$ and these values of x-coordinates called as x-intercepts of graph and the graph intersect y-axis at $y=-4$ called as y-intercept of graph shown in above Figure (1).

Thus, the values of x-intercepts are $-2$, 2 and value of y-intercept is $-4$.

(c)

To determine

To check: The symmetry of graph of the equation $y=x^2-4$.

Answer to Problem 16T

The graph of the equation $y=x^2-4$ is symmetry about y-axis.

Explanation of Solution

The given equation is,

$y=x^2-4$. (1)

Replace y by $-y$ in equation (1) to get new equation,

$-y=x^2-4$

The above equation is not equivalent to equation (1) so the graph is not symmetric about the x-axis.

Replace x by $-x$ in the equation (1) to get the new equation,

$\begin{array}{l}y={\left(-x\right)}^2-4\\ y=x^2-4\end{array}$

The above equation is equivalent to equation (1) so the graph is symmetric about y-axis shown in Figure (1),

The Figure (1) shows that the part of the graph to the left of the y-axis is the mirror image of the part to the right of the y-axis and the points $\left(-2,0\right)$, $\left(2,0\right)$ are reflections of each other about the y-axis.

Thus, the graph of the equation $y=x^2-4$ is symmetry about y-axis.