Find the quadratic function if the vertex is (3,-2) and it passes through the point (4,3). leave your answer in vertex form: f(x)=a(x-h)^2+k

f(x)=

To find the quadratic function in vertex form \( f(x) = a(x-h)^2 + k \), given the vertex and another point on the graph, we can follow these steps:

### Step 1: Identify the given values.
- Vertex: \( (h, k) = (3, -2) \)
- Point: \( (x, y) = (4, 3) \)

### Step 2: Substitute the vertex into the vertex form.
Using the vertex \( (h, k) = (3, -2) \), the function becomes:
\[
f(x) = a(x - 3)^2 - 2
\]

### Step 3: Substitute the point \( (4, 3) \) into the function to find \( a \).
\[
f(4) = 3 = a(4 - 3)^2 - 2
\]
\[
3 = a(1)^2 - 2
\]
\[
3 = a - 2
\]
\[
a = 5
\]

### Step 4: Write the final quadratic function.
\[
f(x) = 5(x - 3)^2 - 2
\]

So, the quadratic function in vertex form is:
\[
f(x) = 5(x - 3)^2 - 2
\]

Would you like further details or have any questions?

Here are 8 related questions you might find interesting:
1. How do you convert a quadratic function from vertex form to standard form?
2. What is the axis of symmetry for the function \( f(x) = 5(x - 3)^2 - 2 \)?
3. How would the graph of the function change if the value of \( a \) was negative?
4. How do you find the x-intercepts of a quadratic function in vertex form?
5. Can a quadratic function have a vertex that is not on the x-axis or y-axis?
6. What are the properties of the vertex form of a quadratic equation?
7. How do you determine the direction in which a parabola opens?
8. How does the value of \( a \) affect the "width" of the parabola?

**Tip:** When dealing with quadratic functions, understanding how the parameters \( a \), \( h \), and \( k \) affect the graph can help you quickly sketch or interpret the function's behavior.

Learn how to find a quadratic function in vertex form given the vertex (3,-2) and passing through the point (4,3). Step-by-step solution with detailed calculations provided.

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