Question

Write the equations of a quadratic function in vertex form with the following information: reflected across the x-axis, vertical stretch of 3, left 9 units, and up 5 units. (1 point) Responses f(x)=−3(x+9)2+5 f of x is equal to negative 3 times open paren x plus 9 close paren squared plus 5 f(x)=−3(x+5)2+9 f of x is equal to negative 3 times open paren x plus 5 close paren squared plus 9 f(x)=3(x+9)2+5 f of x is equal to 3 times open paren x plus 9 close paren squared plus 5 f(x)=−3(x−9)2+5

Answer 1

To form the equation of the quadratic function in vertex form given the specific transformations, let's break down the requirements:

1. Reflected across the x-axis: This means the leading coefficient is negative.
2. Vertical stretch of 3: This increases the absolute value of the leading coefficient to 3.
3. Left 9 units: This indicates a horizontal shift to the left, affecting the x-value in the vertex.
4. Up 5 units: This indicates a vertical shift upward, affecting the y-value in the vertex.

The vertex form of a quadratic function is given by:

\[
f(x) = a(x - h)^2 + k
\]

where \((h, k)\) is the vertex of the parabola and \(a\) controls the direction and the vertical stretch.

### Step 1: Determine the values for \(a\), \(h\), and \(k\)

- Since it is reflected across the x-axis and there is a vertical stretch of 3, \(a\) will be \(-3\).
- The left shift of 9 units means \(h = -9\).
- The upward shift of 5 units means \(k = 5\).

### Step 2: Combine the values into the vertex form equation

Substituting the values into the vertex form:

\[
f(x) = -3(x + 9)^2 + 5
\]

This matches the structure we determined with the transformations specified.

### Conclusion
The correct equation for the quadratic function is:

\[
f(x) = -3(x + 9)^2 + 5
\]

Thus, the answer is:

\[
\boxed{f(x) = -3(x + 9)^2 + 5}
\]