A quadratic function has a vertex at (2,4) and passes through the point (7,-11). Fill in the values of A, B, and C in the equation .

To find the values of \( A \), \( B \), and \( C \) in the quadratic equation of the form:

\[
y = Ax^2 + Bx + C
\]

Given that the vertex is at \( (2, 4) \) and the quadratic passes through the point \( (7, -11) \), we can proceed as follows:

### Step 1: Vertex form of a quadratic function
The vertex form of a quadratic equation is:

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

where \( (h, k) \) is the vertex. Here, \( (h, k) = (2, 4) \). So, the equation becomes:

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

### Step 2: Use the point \( (7, -11) \)
Substitute the point \( (7, -11) \) into the equation to solve for \( A \):

\[
-11 = A(7 - 2)^2 + 4
\]
\[
-11 = A(5)^2 + 4
\]
\[
-11 = 25A + 4
\]
\[
-11 - 4 = 25A
\]
\[
-15 = 25A
\]
\[
A = -\frac{15}{25} = -\frac{3}{5}
\]

### Step 3: Write the quadratic in standard form
Now that we know \( A = -\frac{3}{5} \), we can expand the vertex form into standard form:

\[
y = -\frac{3}{5}(x - 2)^2 + 4
\]

Expand \( (x - 2)^2 \):

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

Now substitute this into the equation:

\[
y = -\frac{3}{5}(x^2 - 4x + 4) + 4
\]
\[
y = -\frac{3}{5}x^2 + \frac{12}{5}x - \frac{12}{5} + 4
\]
\[
y = -\frac{3}{5}x^2 + \frac{12}{5}x + \left(4 - \frac{12}{5}\right)
\]
\[
y = -\frac{3}{5}x^2 + \frac{12}{5}x + \frac{20}{5} - \frac{12}{5}
\]
\[
y = -\frac{3}{5}x^2 + \frac{12}{5}x + \frac{8}{5}
\]

### Step 4: Identify \( A \), \( B \), and \( C \)
From the standard form \( y = Ax^2 + Bx + C \), we have:

- \( A = -\frac{3}{5} \)
- \( B = \frac{12}{5} \)
- \( C = \frac{8}{5} \)

Let me know if you'd like more details or have any questions! 

### 5 Related Questions:
1. How do you convert a quadratic function from vertex form to standard form?
2. What role does the vertex play in the graph of a quadratic function?
3. How can you find the axis of symmetry for a quadratic function?
4. What is the significance of the coefficient \( A \) in a quadratic equation?
5. How can you find the roots of a quadratic function?

### Tip:
The sign of the coefficient \( A \) determines whether the parabola opens upward (positive \( A \)) or downward (negative \( A \)).

A quadratic function has a vertex at (2,4) and passes through the point (7,-11). Fill in the values of A, B, and C in the equation.

Learn how to determine the values of A, B, and C for a quadratic equation with a vertex at (2, 4) and passing through the point (7, -11). This problem covers concepts like the vertex form and standard form of a quadratic equation, including step-by-step algebraic solutions. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation