Q31. - Algebra 2

Question

Complete parts a-c for each quadratic function.

1. Find y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex.
2. Make a table of values that includes the vertex.
3. Use this information to graph the function.

$f\left(x\right)=x^2-\frac{2}{3}x-\frac{8}{9}$

Step-by-Step Solution

Answer

          a. The y-intercept is -0.889, equation of axis of symmetry is $x=0.333$ and x-coordinate of vertex is 0.333.

         b. The table is:

Here, the vertex is $\left(0.333,-1\right)$

     c. The graph is:

a Step 1. Use the concept.

Consider the function $f\left(x\right)=a{x}^2+bx+c,a\ne 0$

• The y-intercept is $f\left(x\right)=a{\left(0\right)}^2+b\left(0\right)+c\text{ or }c$
• The equation of axis of symmetry is $x=\frac{-b}{2a}$
• The x-coordinate of vertex is $\frac{-b}{2a}$

Step 2. Given Information.

The given function is $f\left(x\right)=x^2-\frac{2}{3}x-\frac{8}{9}$

Step 3. Solution.

In the function

$f\left(x\right)=x^2-\frac{2}{3}x-\frac{8}{9}=f\left(x\right)=x^2-0.667x-0.889$, 

we have $a=1,b=-0.667,c=-8.889$

So, the y-intercept is -0.889

The equation of axis of symmetry is given by $x=\frac{-b}{2a}$

Substitute the values $a=1,b=-0.667$ to get:

 $\begin{array}{l}x=\frac{-(-0.667)}{2\left(1\right)}\mathrm{...............}\left[a=1,b=-0.667\right]\\ \Rightarrow x=0.333\end{array}$

So, the equation of axis of symmetry $x=0.333$. Therefore, the x-coordinate of vertex is 0.333.

b Step 1. Use the concept.

Consider the function $f\left(x\right)=a{x}^2+bx+c,a\ne 0$

• The y-intercept is $f\left(x\right)=a{\left(0\right)}^2+b\left(0\right)+c\text{ or }c$
• The equation of axis of symmetry is $x=\frac{-b}{2a}$
• The x-coordinate of vertex is $\frac{-b}{2a}$

Step 2. Given Information.

The given function is $f\left(x\right)=x^2-\frac{2}{3}x-\frac{8}{9}=x^2-0.667x-0.889$ .

From part (a), we have y-intercept is -0.889, equation of axis of symmetry is $x=0.333$and x-coordinate of vertex is 0.333.

Step 3. Discussion.

Choose some values for x that are less than 0.333 and some that are greater than 0.333. This ensures that points on each side of the axis of symmetry are graphed.

Step 4. Table.

Here, the vertex is $\left(0.333,-1\right)$

c Step 1. Use the concept.

The graph of  $f\left(x\right)=a{x}^2+bx+c,a\ne 0$

• The y-intercept is $f\left(x\right)=a{\left(0\right)}^2+b\left(0\right)+c\text{ or }c$
• The equation of axis of symmetry is  $x=\frac{-b}{2a}$
• The x-coordinate of vertex is  $\frac{-b}{2a}$

Step 2. Given Information.

The given function is $f\left(x\right)=x^2-0.667x-0.889$

From part (a) and (b), we have 

Here, the vertex is $\left(0.333,-1\right)$, the y-intercept is -0.889, equation of axis of symmetry is$x=0.333$.

Step 3. Solution.

Graph the vertex and y-intercept. Then graph the points from your table connecting them and the y-intercept with a smooth curve. As a check, draw the axis of symmetry, $x=0.333$, as a green line. The graph of the function should be symmetrical about this line.