what is the equation of a quadratic in standard form that has a vertex at x=4 and an x-intercept at (2,0)

To find the equation of a quadratic in standard form that has a vertex at \(x = 4\) and an x-intercept at \((2, 0)\), we can follow these steps:

### Step 1: Vertex Form of the Quadratic Equation
The vertex form of a quadratic equation is:
\[
y = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex of the parabola. Given that the vertex is at \(x = 4\), we have \(h = 4\). We don’t know \(k\) yet, so the equation becomes:
\[
y = a(x - 4)^2 + k
\]

### Step 2: X-Intercept Condition
An x-intercept occurs where \(y = 0\). Given that the x-intercept is at \((2, 0)\), substitute \(x = 2\) and \(y = 0\) into the equation:
\[
0 = a(2 - 4)^2 + k
\]
Simplifying, we get:
\[
0 = a(-2)^2 + k
\]
\[
0 = 4a + k
\]

### Step 3: Express \(k\) in Terms of \(a\)
From the equation \(0 = 4a + k\), we can express \(k\) as:
\[
k = -4a
\]
Substituting \(k = -4a\) back into the vertex form of the equation, we get:
\[
y = a(x - 4)^2 - 4a
\]

### Step 4: Use the X-Intercept to Find \(a\)
We need another x-intercept to solve for \(a\), but since we are only given one intercept, we assume \(a\) as a constant and rewrite the equation in standard form.

### Step 5: Expand and Simplify to Standard Form
Expand the vertex form to get the standard form of the equation:
\[
y = a(x^2 - 8x + 16) - 4a
\]
Distribute \(a\):
\[
y = ax^2 - 8ax + 16a - 4a
\]
Simplify:
\[
y = ax^2 - 8ax + 12a
\]

### Final Answer:
The quadratic equation in standard form is:
\[
y = ax^2 - 8ax + 12a
\]
Where \(a\) is a constant that you can determine with additional information, such as another point on the parabola.

Would you like more details or have any questions?

Here are some related questions you might consider:
1. How would the equation change if the vertex were at \(x = 4\) and the y-intercept were given?
2. What if the parabola opens downward instead of upward?
3. How can you find the value of \(a\) if another point on the parabola is known?
4. What is the significance of the coefficient \(a\) in the shape of the parabola?
5. How would the equation change if both x-intercepts were given?

**Tip:** In the standard form \(y = ax^2 + bx + c\), the sign of \(a\) determines whether the parabola opens upwards (\(a > 0\)) or downwards (\(a < 0\)).

Learn how to find the equation of a quadratic in standard form with a vertex at x=4 and an x-intercept at (2,0). Step-by-step solution provided, covering vertex form, standard form, and key concepts in quadratic equations. Suitable for high school students.

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