https://file.aiquickdraw.com/mathbot/file/1727930138321bl71vik0.jpg

The graph shows a quadratic function with two key points labeled: \( (2, -6) \) and \( (5, 3) \). From the shape of the graph, we know that the quadratic function is opening upwards, and we can use these points to find the equation of the function.

### General form of a quadratic equation:
\[
f(x) = ax^2 + bx + c
\]
We will use the given points to find \(a\), \(b\), and \(c\). Let's proceed step by step.

### Step 1: Use the vertex form of a quadratic equation
Since one of the points given is the vertex, \((2, -6)\), we can use the vertex form of a quadratic equation:
\[
f(x) = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex, which in this case is \((2, -6)\). So the equation becomes:
\[
f(x) = a(x - 2)^2 - 6
\]

### Step 2: Use the point \((5, 3)\) to find \(a\)
Substitute \(x = 5\) and \(f(x) = 3\) into the equation:
\[
3 = a(5 - 2)^2 - 6
\]
\[
3 = a(3)^2 - 6
\]
\[
3 = 9a - 6
\]
Now, solve for \(a\):
\[
9a = 9 \quad \Rightarrow \quad a = 1
\]

### Step 3: Write the equation in standard form
Since \(a = 1\), the equation becomes:
\[
f(x) = (x - 2)^2 - 6
\]
Now expand this to get the standard form:
\[
f(x) = (x^2 - 4x + 4) - 6
\]
\[
f(x) = x^2 - 4x - 2
\]

Thus, the equation of the quadratic function in standard form is:
\[
f(x) = x^2 - 4x - 2
\]

Would you like further details on any of the steps or a breakdown of the process? Here are five related questions to explore this concept further:

1. How do you convert a quadratic equation from vertex form to standard form?
2. What does the vertex of a quadratic function tell us about its graph?
3. How can you find the roots of a quadratic equation once it’s in standard form?
4. How does the value of \(a\) affect the shape of a quadratic graph?
5. Can you solve this quadratic function using another point on the graph?

**Tip:** When given the vertex and another point, always start with the vertex form of the quadratic equation—it simplifies finding the function's equation.

The graph represents a quadratic function. Write an equation of the function in standard form.

Learn how to find the equation of a quadratic function from a graph with given points. This problem covers key algebraic concepts such as the standard and vertex forms of quadratic equations. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation