Select the graph that represents the function $$ -x^{2}+6 x+27 $$

Question

UpStudy AI · Accepted Answer

1. Write the function as
   $$
   y=-x^{2}+6x+27.
   $$

2. Factor out the coefficient of $$ x^2 $$ from the first two terms:
   $$
   y = -\left(x^2-6x\right) + 27.
   $$

3. Complete the square inside the parentheses. Recall
   $$
   x^2-6x = \left(x-3\right)^2 - 9,
   $$
   so substitute to obtain:
   $$
   y = -\left[\left(x-3\right)^2-9\right] + 27.
   $$

4. Distribute the negative sign:
   $$
   y = -\left(x-3\right)^2 + 9 + 27.
   $$

5. Simplify:
   $$
   y = -\left(x-3\right)^2 + 36.
   $$

6. The vertex form of a parabola is $$ y = a\left(x-h\right)^2 + k $$. Thus, the vertex is at
   $$
   (h,k) = (3,36).
   $$

7. Since the coefficient $$ a=-1 $$ is negative, the parabola opens downward.

8. Find the $$ x $$-intercepts by setting $$ y=0 $$:
   $$
   -\left(x-3\right)^2 + 36 = 0 \quad \Rightarrow \quad \left(x-3\right)^2 = 36.
   $$
   Taking the square root of both sides gives
   $$
   x-3 = \pm 6,
   $$
   so the solutions are
   $$
   x = 3+6=9 \quad \text{and} \quad x = 3-6=-3.
   $$

9. The $$ y $$-intercept is found by setting $$ x=0 $$:
   $$
   y = -0^{2}+6(0)+27 = 27.
   $$

In summary, the graph of $$ y=-x^{2}+6x+27 $$ is a downward opening parabola with vertex $$ (3,36) $$, x-intercepts $$ (-3,0) $$ and $$ (9,0) $$, and y-intercept $$ (0,27) $$.

Thus, the correct graph is the one with vertex at $$ (3,36) $$, symmetric about the line $$ x=3 $$, that opens downward and passes through the points $$ (-3,0) $$, $$ (9,0) $$, and $$ (0,27) $$.
The graph of $$ y = -x^{2} + 6x + 27 $$ is a downward-opening parabola with its vertex at $$ (3, 36) $$. It crosses the x-axis at $$ (-3, 0) $$ and $$ (9, 0) $$, and the y-axis at $$ (0, 27) $$.

Select the graph that represents the function

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Algebra Questions