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Chapter 3: Problem 21

Find the vertex of the graph of each quadratic function. $$f(x)=3 x^{2}-12 x+1$$

Short Answer

Expert verified
The vertex is (2, -11).

Step by step solution

01

Identify the coefficients

For the quadratic function in the form of \( f(x) = ax^2 + bx + c \), identify the coefficients. Here, \( a = 3 \), \( b = -12 \), and \( c = 1 \)
02

Use the vertex formula

The x-coordinate of the vertex of a quadratic function is given by the formula \( x = \frac{-b}{2a} \). Substitute the coefficients into the formula. \( x = \frac{-(-12)}{2 \cdot 3} = \frac{12}{6} = 2 \).
03

Find the y-coordinate

Substitute the x-coordinate back into the original function to find the y-coordinate. \( f(2) = 3(2)^2 - 12(2) + 1 = 3 \cdot 4 - 24 + 1 = 12 - 24 + 1 = -11 \).
04

Write the vertex

Combine the x-coordinate and y-coordinate as an ordered pair. The vertex is \( (2, -11) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Functions
A quadratic function is a type of polynomial function that takes the form: \(f(x) = ax^2 + bx + c\). These functions graph as parabolas, which are U-shaped curves.
Quadratic functions have several key properties, such as:
  • They can open upwards or downwards depending on the coefficient \(a\).
  • The highest or lowest point on the graph is called the vertex.
  • They possess a symmetrical axis that goes through the vertex.
Understanding these properties helps us analyze and solve quadratic functions more easily.
Vertex Formula
The vertex formula is crucial for finding the vertex of a quadratic function.
The general form of the vertex formula for the x-coordinate of the vertex is given by: \( x = \frac{-b}{2a}\)
To use this formula, you:
  • Identify the coefficients \(a\) and \(b\) of the quadratic function.
  • Substitute these values into the vertex formula.
This gives you the x-coordinate of the vertex, \(x_v\). To find the y-coordinate \(y_v\), substitute \(x_v\) back into the original quadratic equation.
This process allows you to derive the vertex as an ordered pair \((x_v, y_v)\).
Coefficients of a Quadratic Function
The coefficients of a quadratic function in the standard form \(f(x) = ax^2 + bx + c\) play significant roles in determining the properties of the graph.
Here,
  • \(a\) is the coefficient of \(x^2\),
  • \(b\) is the coefficient of \(x\), and
  • \(c\) is the constant term.

Key points include:
  • \(a\) determines the direction of the parabola (upward if \(a > 0\), downward if \(a < 0\)).
  • \(b\) influences the vertex's position horizontally on the graph.
Understanding these coefficients helps in applying the vertex formula correctly.
X-Coordinate and Y-Coordinate
Finding the x-coordinate and y-coordinate of the vertex involves simple steps.
First, use the vertex formula to find the x-coordinate: \(x = \frac{-b}{2a}\).
Next, substitute the x-coordinate back into the quadratic function to find the corresponding y-coordinate, \(f(x)\). For example, for the function \(f(x) = 3x^2 - 12x + 1\):
1. Identify coefficients: \(a = 3\), \(b = -12\).
2. Apply vertex formula: \(x = \frac{-(-12)}{2*3} = 2\).
3. Find y-coordinate: \(f(2) = 3(2^2) - 12(2) + 1 = -11\).
So, the vertex is at \((2, -11)\).

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