A parabola opening up or down has vertex (4,0) and passes through 17,
–
169
20
. Write its equation in vertex form

The vertex form of a parabola is given by:

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

where \((h, k)\) is the vertex of the parabola, and \(a\) is a constant that determines the "stretch" or "compression" of the parabola.

We are given:
- The vertex \((h, k) = (4, 0)\).
- A point \(\left( 17, -\frac{169}{20} \right)\) through which the parabola passes.

We can substitute these values into the vertex form equation to find the value of \(a\).

### Step 1: Substitute the vertex into the equation
Since \((h, k) = (4, 0)\), the equation becomes:

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

### Step 2: Substitute the point \(\left( 17, -\frac{169}{20} \right)\) into the equation to solve for \(a\)

Substitute \(x = 17\) and \(y = -\frac{169}{20}\):

\[
-\frac{169}{20} = a(17 - 4)^2
\]

\[
-\frac{169}{20} = a(13)^2
\]

\[
-\frac{169}{20} = a(169)
\]

### Step 3: Solve for \(a\)

\[
a = \frac{-\frac{169}{20}}{169} = \frac{-169}{20 \times 169} = \frac{-1}{20}
\]

Thus, \(a = -\frac{1}{20}\).

### Step 4: Write the final equation

The equation of the parabola is:

\[
y = -\frac{1}{20}(x - 4)^2
\]

This is the equation of the parabola in vertex form.

---

Would you like more details, or do you have any questions about the process?

Here are 5 related questions to expand on the concept:

1. How would the equation change if the parabola opens to the left or right?
2. What is the significance of the value of \(a\) in the vertex form?
3. How do we derive the general form of a parabola from its vertex form?
4. Can you find the focus and directrix of this parabola?
5. How does the graph change if the vertex is shifted to a different location?

**Tip**: The value of \(a\) determines the direction and width of the parabola. A negative \(a\) indicates that the parabola opens downward.

A parabola opening up or down has vertex (4,0) and passes through (17, -169/20). Write its equation in vertex form.

Learn how to find the equation of a parabola that has vertex (4,0) and passes through the point (17, -169/20). This step-by-step solution explains how to apply the vertex form of a quadratic equation to determine the value of 'a' and the final equation. Ideal for students in Grades 9-12.

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