Chapter 8.3, Problem 36E

To determine

To calculate: The vertex of the graph of the quadratic function $y=0.25x^2-5^{\frac{2}{3}}x+2$ .

Answer to Problem 36E

The vertex of the graph of the quadratic function $y=0.25x^2-5^{\frac{2}{3}}x+2$ is $\left(5.848,-6.55\right)$ .

Explanation of Solution

Given information:

The quadratic function $y=0.25x^2-5^{\frac{2}{3}}x+2$ .

Formula used:

The quadratic function of the form $y=a{x}^2+bx+c$ , the y-coordinate of the vertex is the maximum value of the function when $a<0$ and minimum value of the function when $a>0$ .

Calculation:

Consider the quadratic function $y=0.25x^2-5^{\frac{2}{3}}x+2$ .

The steps to evaluate the vertex of the function is provided below,

Step 1: Press $\boxed{\text{Y}=}$ key.

Step 2: Enter the function $y=0.25x^2-5^{\frac{2}{3}}x+2$ then press the $\boxed{\text{WINDOW}}$ key to change the scaling of x-axis to $-5\text{ to }5$ .

Step 3: Press $\boxed{\text{GRAPH}}$ key.

The result obtained is provided below,

  

Now, the graph open upward so the function has minimum value. Also recall that quadratic function of the form $y=a{x}^2+bx+c$ , the y-coordinate of the vertex is the maximum value of the function when $a<0$ and minimum value of the function when $a>0$ .

Here, $a=0.25$ .

Step 4: Press the $\boxed{\text{2nd}}$ key and then $\boxed{\text{TRACE}}$ key.

Step 5: Choose the third option that is minimum from the list.

Step 6: Enter the left bound as $0$ , right bound as $1.021$ and guess as $0.468$ where the vertex might lie with help of arrow key and press $\boxed{\text{ENTER}}$ key.

The result obtained is provided below,

  

The vertex of the graph of the quadratic function $y=0.25x^2-5^{\frac{2}{3}}x+2$ is $\left(5.848,-6.55\right)$ .