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Chapter 3: Problem 23

Find the vertex of the graph of each quadratic function. $$f(x)=-3(x-4)^{2}+1$$

Short Answer

Expert verified
The vertex is \( (4, 1) \).

Step by step solution

01

Identify the standard vertex form

The given quadratic function is in the vertex form, which is \[ f(x) = a(x-h)^2 + k \] where \(a\) is the coefficient of the squared term, \(h\) is the x-coordinate, and \(k\) is the y-coordinate of the vertex.
02

Extract the vertex coordinates

Compare the given function \[ f(x) = -3(x-4)^2 + 1 \] with the standard vertex form. Identify that\( h = 4 \) and \( k = 1 \).
03

State the vertex

Since \( h = 4 \) and \( k = 1 \), the vertex of the given quadratic function is the point \( (4, 1) \).

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Key Concepts

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

quadratic function
A quadratic function is a type of polynomial function that features a variable raised to the second power as its highest degree. The general form of a quadratic function is given by:

\[ f(x) = ax^2 + bx + c \] where:
  • a is the coefficient of the squared term, also known as the leading coefficient
  • b is the coefficient of the linear term
  • c is the constant term
Quadratic functions create parabolas when graphed. The direction in which the parabola opens (upwards or downwards) depends on the sign of a:
  • If a is positive, the parabola opens upwards.
  • If a is negative, the parabola opens downwards.
Understanding the shape and position of a quadratic graph is critical for many algebra problems, especially when identifying key features like the vertex, axis of symmetry, and intercepts.
vertex form
The vertex form of a quadratic function is especially useful for identifying the vertex of the parabola. A quadratic function in vertex form is written as:

\[ f(x) = a(x-h)^2 + k \]where:
  • a is the same leading coefficient as in the standard form, dictating the width and direction of the parabola
  • h is the x-coordinate of the vertex
  • k is the y-coordinate of the vertex
Using the vertex form makes it easier to quickly identify the vertex coordinates \((h, k)\). For example, consider the function given in the exercise: \[ f(x) = -3(x-4)^2 + 1 \]Here, the vertex can be identified directly from the form of the function:
  • h is 4, representing the x-coordinate of the vertex
  • k is 1, representing the y-coordinate of the vertex
identifying vertex
Identifying the vertex of a quadratic function in vertex form is straightforward and involves matching the function to the template \((h, k)\). For the exercise function:

\[ f(x) = -3(x-4)^2 + 1 \],you can see that:
  • x-h translates to \(x-4\), so h is 4.
  • The constant term added at the end is 1, so k is 1.
Thus, the vertex of the quadratic function is at the point \((4, 1)\). This systematic approach allows anyone to find the vertex accurately and easily. When you recognize the vertex form, you can simply extract and state the vertex coordinates, which are critical in understanding the function's graph and behavior.

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