Write an equation of the quadratic function with the vertex at the point (-1, 4) and such that f (0) = 6.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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### Mathematics Exercise: Inverse Functions

**Problem Statement:**

1. Given the function \( f(x) = \frac{x^3}{2} + 4 \) with the domain \( 1 \leq x \leq 3 \):

   a) Find the formula for the inverse function \( f^{-1}(x) \).

   b) Determine the domain of \( f^{-1}(x) \).

---

For visual aids accompanying this problem, consider diagrams illustrating the function \( f(x) \) and its inverse \( f^{-1}(x) \). These diagrams can include:

- **Graph of \( f(x) \):** A cubic graph opening to the right, showing the range for \( x \) within [1, 3]. 

- **Graph of \( f^{-1}(x) \):** A reflection of the graph of \( f(x) \) over the line \( y = x \), illustrating the domain and range swap for the inverse function.

These visualizations can help in understanding the transformation from the original function to its inverse.

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Transcribed Image Text:Certainly! Here's the transcribed text tailored for an educational website: --- ### Mathematics Exercise: Inverse Functions **Problem Statement:** 1. Given the function \( f(x) = \frac{x^3}{2} + 4 \) with the domain \( 1 \leq x \leq 3 \): a) Find the formula for the inverse function \( f^{-1}(x) \). b) Determine the domain of \( f^{-1}(x) \). --- For visual aids accompanying this problem, consider diagrams illustrating the function \( f(x) \) and its inverse \( f^{-1}(x) \). These diagrams can include: - **Graph of \( f(x) \):** A cubic graph opening to the right, showing the range for \( x \) within [1, 3]. - **Graph of \( f^{-1}(x) \):** A reflection of the graph of \( f(x) \) over the line \( y = x \), illustrating the domain and range swap for the inverse function. These visualizations can help in understanding the transformation from the original function to its inverse. ---
**Problem Statement:**

Write an equation of the quadratic function with the vertex at the point \((-1, 4)\) and such that \(f(0) = 6\).

---

**Solution:**

To find the equation of a quadratic function given its vertex, we use the vertex form of a quadratic equation:

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

Where \( (h, k) \) is the vertex of the parabola. Given the vertex \((-1, 4)\), the equation becomes:

\[ f(x) = a(x + 1)^2 + 4 \]

We also know that \( f(0) = 6 \). Substitute \( x = 0 \) into the equation to find \( a \):

\[ 6 = a(0 + 1)^2 + 4 \]

\[ 6 = a(1) + 4 \]

\[ 2 = a \]

Thus, the equation of the quadratic function is:

\[ f(x) = 2(x + 1)^2 + 4 \]
Transcribed Image Text:**Problem Statement:** Write an equation of the quadratic function with the vertex at the point \((-1, 4)\) and such that \(f(0) = 6\). --- **Solution:** To find the equation of a quadratic function given its vertex, we use the vertex form of a quadratic equation: \[ f(x) = a(x - h)^2 + k \] Where \( (h, k) \) is the vertex of the parabola. Given the vertex \((-1, 4)\), the equation becomes: \[ f(x) = a(x + 1)^2 + 4 \] We also know that \( f(0) = 6 \). Substitute \( x = 0 \) into the equation to find \( a \): \[ 6 = a(0 + 1)^2 + 4 \] \[ 6 = a(1) + 4 \] \[ 2 = a \] Thus, the equation of the quadratic function is: \[ f(x) = 2(x + 1)^2 + 4 \]
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