Write the standard form of the quadratic function that has the indicated vertex and whose graph passes through the given point. Use a graphing utility to verify your result.
Vertex: 
(−4, −4);
    Point: 
(−2, 1)

To find the standard form of a quadratic function given its vertex and a point, we use the **vertex form** of a quadratic function, which is:

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

where \( (h, k) \) is the vertex and \( a \) is a constant. In this case, the vertex is \( (-4, -4) \), so the function becomes:

\[
f(x) = a(x + 4)^2 - 4
\]

Next, we use the given point \( (-2, 1) \) to find the value of \( a \). Substituting \( x = -2 \) and \( f(x) = 1 \) into the equation:

\[
1 = a(-2 + 4)^2 - 4
\]
\[
1 = a(2)^2 - 4
\]
\[
1 = 4a - 4
\]
\[
5 = 4a
\]
\[
a = \frac{5}{4}
\]

Now that we have \( a = \frac{5}{4} \), we can write the quadratic function in vertex form:

\[
f(x) = \frac{5}{4}(x + 4)^2 - 4
\]

To convert this into standard form, expand the equation:

\[
f(x) = \frac{5}{4}(x^2 + 8x + 16) - 4
\]
\[
f(x) = \frac{5}{4}x^2 + \frac{40}{4}x + \frac{80}{4} - 4
\]
\[
f(x) = \frac{5}{4}x^2 + 10x + 20 - 4
\]
\[
f(x) = \frac{5}{4}x^2 + 10x + 16
\]

Thus, the quadratic function in standard form is:

\[
f(x) = \frac{5}{4}x^2 + 10x + 16
\]

### Graph Verification
To verify this, you can use a graphing utility to plot both the function and check if the vertex is at \( (-4, -4) \) and if the function passes through the point \( (-2, 1) \).

Would you like further details on the graph verification process?

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Here are 5 related questions for practice:
1. How do you find the vertex of a quadratic function in standard form?
2. How does the value of \( a \) affect the direction of the parabola?
3. What are the steps to convert a quadratic function from vertex form to standard form?
4. How do you find the axis of symmetry for a quadratic function?
5. How do you determine whether a quadratic function has a maximum or minimum value?

**Tip:** The vertex form of a quadratic equation is useful when you already know the vertex, as it makes finding the equation more straightforward.

Write the standard form of the quadratic function that has the indicated vertex and whose graph passes through the given point. Use a graphing utility to verify your result. Vertex: (−4, −4); Point: (−2, 1)

Learn how to write the standard form of a quadratic function with the vertex (-4, -4) that passes through the point (-2, 1). This solution covers key concepts in algebra, including vertex form and converting to standard form, with a graph verification step. Suitable for Grades 9-11 students.

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