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Chapter 9: Problem 11

Identify the vertex of each parabola. $$ f(x)=(x-1)^{2} $$

Short Answer

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Step by step solution

01

Identify the vertex form of a parabola

The function given is: \( f(x) = (x-1)^2 \). This is a quadratic function in vertex form, which is generally \( f(x) = a(x-h)^2 + k \).
02

Determine the vertex coordinates

In the vertex form \( f(x) = a(x-h)^2 + k \), the vertex of the parabola is given by the coordinates \( (h, k) \). Compare the given function to the vertex form: \( f(x) = (x-1)^2 \). Here, \( h = 1 \) and \( k = 0 \). Thus, the vertex is at \( (1, 0) \).

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Key Concepts

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

parabola
A parabola is a U-shaped curve that represents the graph of a quadratic function. It can open either upwards or downwards, depending on the coefficient of the squared term. The shape and position of the parabola are influenced by its equation.

Key features of a parabola include:
  • Vertex: The highest or lowest point of the parabola.
  • Axis of Symmetry: A vertical line that divides the parabola into two symmetrical halves.
  • Focus and Directrix: Point and line used in the geometric definition of a parabola, but not always needed for basic graphing.
In the exercise provided, we'll focus on identifying the vertex, the main point associated with locating a parabola on a graph.
vertex form
The vertex form of a quadratic function makes it easy to identify the vertex of a parabola. This form is represented as:

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

In this equation:
  • represents the x-coordinate of the vertex.
  • \( k \) represents the y-coordinate of the vertex.
  • \( a \) affects the width and direction of the parabola (whether it opens upwards if \( a > 0 \) or downwards if \( a < 0 \)).
For the given function \( f(x) = (x-1)^2 \), you can compare it to the general form. Here, \( h = 1 \) and \( k = 0 \), placing the vertex at \( (1, 0) \). It's a straightforward process once you know how to recognize the vertex form.
quadratic function
A quadratic function is any function that can be written in the form:

\( f(x) = ax^2 + bx + c \)

Where \( a, b, \) and \( c \) are constants, and \( a \) is not zero. The graph of a quadratic function is always a parabola. Depending on the values of \( a, b, \) and \( c \), the parabola might be shifted or flipped in various ways.

Key characteristics of quadratic functions include:
  • The vertex (turning point of the parabola).
  • The direction in which the parabola opens (determined by the sign of \( a \)).
  • The axis of symmetry which is the vertical line that passes through the vertex, given by the equation \( x = h \).
  • The roots (or zeros) where the function intersects the x-axis.
Understanding these aspects is essential for working with quadratic functions effectively.

In the given example \( f(x) = (x-1)^2 \), it's already presented in vertex form, making it simpler to identify the vertex and plot the function.

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Most popular questions from this chapter

The snow depth in a particular location varies throughout the winter. In a typical winter, the snow depth in inches might be approximated by the following function. $$ f(x)=\left\\{\begin{array}{ll} 6.5 x & \text { if } 0 \leq x \leq 4 \\ -5.5 x+48 & \text { if } 4

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Graph each absolute value function. \(f(x)=|2-x|\)

Use the tests for symmetry to decide whether the graph of each relation is symmetric with respect to the \(x\) -axis, the y-axis, or the origin. More than one of these symmetries, or none of them, may apply. $$ -2 x=y $$
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