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Chapter 8: Problem 1

Sketch the graph of each quadratic function. Label the vertex and sketch and label the axis of symmetry. See Examples I through 5 . $$ f(x)=x^{2}-1 $$

Short Answer

Expert verified
The vertex is at (0, -1), and the axis of symmetry is x = 0.

Step by step solution

01

Identify the Function Form

The function is given as \( f(x) = x^2 - 1 \). This is a quadratic function in the standard form \( y = ax^2 + bx + c \), where \( a = 1 \), \( b = 0 \), and \( c = -1 \).
02

Find the Vertex

The vertex of a quadratic function in the form \( f(x) = ax^2 + bx + c \) can be found using the formula \( x = \frac{-b}{2a} \). Substituting \( b = 0 \) and \( a = 1 \), we get \( x = 0 \). To find the y-coordinate, substitute \( x = 0 \) into the function: \( f(0) = 0^2 - 1 = -1 \). Thus, the vertex is at the point \( (0, -1) \).
03

Sketch the Graph

The graph of \( f(x) = x^2 - 1 \) is a parabola that opens upwards (since \( a = 1 > 0 \)). The vertex of the parabola is at \( (0, -1) \). Plot this point, and draw a U-shaped curve passing through the vertex, opening upwards.
04

Axis of Symmetry

The axis of symmetry of a quadratic function \( y = ax^2 + bx + c \) is the vertical line that passes through the vertex. For this function, the axis of symmetry is \( x = 0 \). Draw a dashed vertical line through \( x = 0 \) to represent the axis of symmetry.
05

Label the Graph

Label the vertex \( (0, -1) \) on the graph, and label the axis of symmetry as \( x = 0 \).

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Key Concepts

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

Vertex of a Quadratic Function
The vertex of a quadratic function is a key feature that determines its position on the graph. In the quadratic equation \[ f(x) = ax^2 + bx + c \], the vertex is the point \( (h, k) \) where the function reaches its minimum or maximum value. The location of the vertex can be found using the formula: \[ x = \frac{-b}{2a} \].
For the function \( f(x) = x^2 - 1 \), both \( b \) and \( a \) have been identified leading to \( x = \frac{-0}{2 \times 1} = 0 \).
Plugging in \( x = 0 \)back into the function, we find:
  • \( f(0) = 0^2 - 1 = -1 \)
Thus, the vertex is at the coordinate \( (0, -1) \).
The vertex can inform us about the quadratic's nature:
  • It's a minimum point if \( a > 0 \) (u-shaped parabola).
  • It's a maximum point if \( a < 0 \) (upsidedown u-shaped parabola).
Axis of Symmetry
The axis of symmetry is an imaginary line that provides balance to the parabola, splitting it into two mirror-image halves. Understanding this concept is crucial for accurate parabola sketching. For any quadratic function \[ y = ax^2 + bx + c \], the axis of symmetry can be located using the same formula used to find the x-coordinate of the vertex:\[ x = \frac{-b}{2a} \].
For our example, \( f(x) = x^2 - 1 \), plugging in the values gives us:
  • \( x = \frac{0}{2(1)} = 0 \)
Thus, the axis of symmetry is the vertical line \( x = 0 \).
Here鈥檚 how to visualize it:
  • It goes vertically down through the vertex.
  • It ensures each side of the parabola is a mirror image of the other.
  • Always represents symmetry in the parabola's shape.
Parabola Graphing
Graphing a parabola involves understanding its shape and direction: does it open upwards or downwards? Here, we鈥檙e working with \( f(x) = x^2 - 1 \). Since \( a = 1 \) (positive), the parabola opens upwards. This is a fundamental quality that affects how you sketch it.
To effectively sketch the parabola:
  • Start by plotting the vertex at \((0, -1)\).
  • Use the axis of symmetry \( x = 0 \) to balance your sketch.
  • Draw the u-shaped curve starting from the vertex and extending upwards.
When graphing, keep these key points in mind:
  • Parabolas have a smooth, continuous curve.
  • The distance from any point on the parabola to the axis of symmetry is equal at any given \( y \).
  • The vertex is a crucial guidepost for the overall graph shape.

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