To approximate the zeros and find the vertex of the function [tex]\( y = -x^2 + 8x + 10 \)[/tex] using a graphing calculator, follow these steps:
### Step-by-Step Solution:
1. Identify the function to graph:
[tex]\[
y = -x^2 + 8x + 10
\][/tex]
2. Vertex Calculation:
The vertex form of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] is given by:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
Substituting [tex]\( a = -1 \)[/tex] and [tex]\( b = 8 \)[/tex], we get:
[tex]\[
x = -\frac{8}{2 \cdot -1} = 4
\][/tex]
To find the y-coordinate of the vertex, substitute [tex]\( x = 4 \)[/tex] back into the function:
[tex]\[
y = -4^2 + 8(4) + 10 = -16 + 32 + 10 = 26
\][/tex]
Therefore, the vertex is [tex]\( (4, 26) \)[/tex].
3. Calculate the zeros of the function:
To find the zeros, set [tex]\( y \)[/tex] to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
0 = -x^2 + 8x + 10
\][/tex]
Solving this quadratic equation, we get:
[tex]\[
-x^2 + 8x + 10 = 0
\][/tex]
Rearrange the equation to:
[tex]\[
x^2 - 8x - 10 = 0
\][/tex]
Using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex] where [tex]\( a = 1 \)[/tex], [tex]\( b = -8 \)[/tex], and [tex]\( c = -10 \)[/tex], we find:
[tex]\[
x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(-10)}}{2(1)}
\][/tex]
[tex]\[
x = \frac{8 \pm \sqrt{64 + 40}}{2}
\][/tex]
[tex]\[
x = \frac{8 \pm \sqrt{104}}{2}
\][/tex]
[tex]\[
x = \frac{8 \pm 2\sqrt{26}}{2}
\][/tex]
[tex]\[
x = 4 \pm \sqrt{26}
\][/tex]
4. Approximate the zeros:
[tex]\[
x_1 = 4 - \sqrt{26}
\][/tex]
[tex]\[
x_2 = 4 + \sqrt{26}
\][/tex]
These approximate values are:
[tex]\[
x_1 \approx 4 - 5.1 = -1.1
\][/tex]
[tex]\[
x_2 \approx 4 + 5.1 = 9.1
\][/tex]
### Conclusion:
From the calculations:
- The vertex of the function [tex]\( y = -x^2 + 8x + 10 \)[/tex] is at [tex]\((4, 26)\)[/tex].
- The zeros of the function are approximately [tex]\((-1.1, 0)\)[/tex] and [tex]\((9.1, 0)\)[/tex].
Therefore, the filled solution is:
[tex]\[
\begin{array}{l}
\qquad y=-x^2+8 x+10 \\
\text { Vertex: ( } 4,26 \text { ) } \\
\text { Solutions: } (-1.1, 0) \text { ) and } (9.1, 0 \text { ) }
\end{array}
\][/tex]
