Chapter 4.1, Problem 21PPS

(A)

To determine

To find:

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

Answer to Problem 21PPS

  $V=\left(6,3\right)$

Explanation of Solution

Given:

The given equation is

  $f\left(x\right)=-2x^2+3x+9$

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)

  $\begin{array}{l}k=a{h}^2+bh+c\\ {\left(y-k\right)}^2=4a\left(x-h\right)\mathrm{.......................}\left(2\right)\\ f\left(x\right)=-2x^2+3x+9\end{array}$

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

The vertex is

  $V=\left(6,3\right)$

(B)

To determine

To show:

The table of the parabola is of the given form

Explanation of Solution

Given:

The equation of parabola

  $f\left(x\right)=-2x^2+3x+9$

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 table

  $f\left(x\right)=-2x^2+3x+9$

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

$x-axis$	$0$	$-1.5$	$3.3$	$3$
$y-axis$	$9$	$0$	$-3$	$0$

(C)

To determine

To show:

The graph of the parabola is of the given form

Explanation of Solution

Given:

The equation of parabola

  $f\left(x\right)=-2x^2+3x+9$

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

  $f\left(x\right)=-2x^2+3x+9$

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