### Finding the Vertex of a Quadratic FunctionFor educational purposes, let's find the vertex of the graph of the quadratic function $ f(x) = 2x^2 - 4x - 1 $.#### Step-by-Step Process1. **Identify the coefficients $ a $ and $ b $**: In the given function $ f(x) = 2x^2 - 4x - 1 $: - $ a = 2 $ - $ b = -4 $2. **Use the vertex formula**: The vertex $ (h, k) $ of a parabola $ f(x) = ax^2 + bx + c $ is given by: \[ h = -\frac{b}{2a} \] Substituting $ a = 2 $ and $ b = -4 $: \[ h = -\frac{-4}{2(2)} \] Simplifying this expression: \[ h = \frac{4}{4} = 1 \] Thus, the x-coordinate of the vertex is $ 1 $.3. **Find the y-coordinate (k) by substituting $ h $ back into the function**: \[ f(h) = f(1) = 2(1)^2 - 4(1) - 1 \] Simplifying within the function: \[ f(1) = 2 - 4 - 1 = -3 \] Hence, the y-coordinate of the vertex is $ -3 $.The vertex of the graph $ f(x) = 2x^2 - 4x - 1 $ is $ (1, -3) $.### Example Problem for PracticeFind the vertex of the graph $ f(x) = 2x^2 - 4x + 4 $.Given the function $ f(x) = 2x^2 - 4x + 4 $:1. Identify values of $ a $ and $ b $: - $ a = 2 $ - $ b = -4 $2. Use the vertex formula to find the x-coordinate ($ h $): \[ h = -\frac{b}{2a} \

See below the calculation and answer

Find the vertex of graph of f(x) = 2x2 – 4x – 1. We will identify the values of a and b. Then we can substitute into the formula for the vertex of a parabola to find its coordinates. The function is written in f(x) = ax² + bx + c form, where a = 2 and b = -4. To find the vertex of its graph, we calculate the following. Substitute 1 Substitute 2 for a and -4 for b. = 2a 2(2) for - 2a -4 =O (1)2 – 4(1) – 1 %3D This is the y-coordinate of the vertex. This is x-coordinate of the vertex. The vertex is the point (1, -3). Find the vertex of the graph of f(x) = 2x² – 4x + 4. (х, у) %3D

Find the vertex of graph of f(x) = 2x2 – 4x – 1. We will identify the values of a and b. Then we can substitute into the formula for the vertex of a parabola to find its coordinates. The function is written in f(x) = ax² + bx + c form, where a = 2 and b = -4. To find the vertex of its graph, we calculate the following. Substitute 1 Substitute 2 for a and -4 for b. = 2a 2(2) for - 2a -4 =O (1)2 – 4(1) – 1 %3D This is the y-coordinate of the vertex. This is x-coordinate of the vertex. The vertex is the point (1, -3). Find the vertex of the graph of f(x) = 2x² – 4x + 4. (х, у) %3D

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

### Finding the Vertex of a Quadratic Function

For educational purposes, let's find the vertex of the graph of the quadratic function $ f(x) = 2x^2 - 4x - 1 $.

#### Step-by-Step Process

1. **Identify the coefficients $ a $ and $ b $**:
In the given function $ f(x) = 2x^2 - 4x - 1 $:
- $ a = 2 $
- $ b = -4 $

2. **Use the vertex formula**:
The vertex $ (h, k) $ of a parabola $ f(x) = ax^2 + bx + c $ is given by:
\[
h = -\frac{b}{2a}
\]
Substituting $ a = 2 $ and $ b = -4 $:
\[
h = -\frac{-4}{2(2)}
\]
Simplifying this expression:
\[
h = \frac{4}{4} = 1
\]
Thus, the x-coordinate of the vertex is $ 1 $.

3. **Find the y-coordinate (k) by substituting $ h $ back into the function**:
\[
f(h) = f(1) = 2(1)^2 - 4(1) - 1
\]
Simplifying within the function:
\[
f(1) = 2 - 4 - 1 = -3
\]
Hence, the y-coordinate of the vertex is $ -3 $.

The vertex of the graph $ f(x) = 2x^2 - 4x - 1 $ is $ (1, -3) $.

### Example Problem for Practice

Find the vertex of the graph $ f(x) = 2x^2 - 4x + 4 $.

Given the function $ f(x) = 2x^2 - 4x + 4 $:

1. Identify values of $ a $ and $ b $:
- $ a = 2 $
- $ b = -4 $

2. Use the vertex formula to find the x-coordinate ($ h $):
\[
h = -\frac{b}{2a}
\

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