Chapter 5.1, Problem 16PPS

(a)

To determine

To find:

The equation of the axis of symmetry, $y$ -intercept and the $x$ -cordinate of the vertex.

Answer to Problem 16PPS

  $V=\left(-0.8,0.8\right)$

Explanation of Solution

Given:

The given equation is

  $f\left(x\right)=4x^2-3$

Concept used:

The standard form of parabola equation

  $y=a{x}^2+bx+c$

Putting $x=h,y=k$ in equation (1)

  $k=a{h}^2+bh+c\mathrm{.......................}\left(2\right)$

Where $\left(h,k\right)$ is the vertex .

Calculation:

The standard form of parabola equation

  $y=a{x}^2+bx+c$

Putting $x=h,y=k$ in equation (1)

  $\begin{array}{l}k=a{h}^2+bh+c\\ {\left(y-k\right)}^2=4a\left(x-h\right)\mathrm{.......................}\left(2\right)\\ f\left(x\right)=4x^2-3\end{array}$

Where $\left(h,k\right)$ is the vertex

The vertex is

  $V=\left(-0.8,0.8\right)$

(b)

To determine

To show:

The table of the parabola is of the given form

Explanation of Solution

Given:

The equation of parabola

  $f\left(x\right)=4x^2-3$

Concept used:

The standard form of parabola equation

  $y=a{x}^2+bx+c$

Putting $x=h,y=k$ in equation (1)

  $k=a{h}^2+bh+c\mathrm{.......................}\left(2\right)$

Where $\left(h,k\right)$ is the vertex

Calculation:

The standard form of parabola equation

  $y=a{x}^2+bx+c$

Putting $x=h,y=k$ in equation (1)

  $\begin{array}{l}k=a{h}^2+bh+c\\ {\left(y-k\right)}^2=4a\left(x-h\right)\mathrm{.......................}\left(2\right)\end{array}$

Where $\left(h,k\right)$ is the vertex .

Draw the table

  $f\left(x\right)=4x^2-3$

Test one point in each of the region formed by the graph .

If the point satisfies the function then shade the entire region to denote that every point in the region satisfies the function .

$x-axis$	$0$	$-0.8$	$0.8$	$-1$
$y-axis$	$-3$	$0$	$0$	$1$

(c)

To determine

To show:

The graph of the parabola is of the given form .

Explanation of Solution

Given:

The equation of parabola

  $f\left(x\right)=4x^2-3$

Concept used:

The stabdard form of parabola equation

  $y=a{x}^2+bx+c$

Putting $x=h,y=k$ in equation (1)

  $k=a{h}^2+bh+c\mathrm{.......................}\left(2\right)$

Where $\left(h,k\right)$ is the vertex

Calculation:

The stabdard form of parabola equation

  $y=a{x}^2+bx+c$

Putting $x=h,y=k$ in equation (1)

  $\begin{array}{l}k=a{h}^2+bh+c\\ {\left(y-k\right)}^2=4a\left(x-h\right)\mathrm{.......................}\left(2\right)\end{array}$

Where $\left(h,k\right)$ is the vertex

Draw the graph

  $f\left(x\right)=4x^2-3$

Test one point in each of the region formed by the graph .

If the point satisfies the function then shade the entire region to denote that every point in the region satisfies the function .