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Chapter 8: Problem 4

Sketch the graph of each quadratic function. Label the vertex and sketch and label the axis of symmetry. See Examples I through 5 . $$ h(x)=x^{2}-4 $$

Short Answer

Expert verified
The vertex is at (0, -4), and the axis of symmetry is the line x = 0.

Step by step solution

01

Identify the Standard Form

The quadratic function given is \( h(x) = x^2 - 4 \). This is in the standard form of a quadratic function, \( y = ax^2 + bx + c \), where \( a = 1 \), \( b = 0 \), and \( c = -4 \).
02

Find the Vertex

For a quadratic function \( y = ax^2 + bx + c \), the vertex can be found using the formula \( x = -\frac{b}{2a} \). Substituting \( a = 1 \) and \( b = 0 \) gives \( x = -\frac{0}{2 \times 1} = 0 \). Substituting this value back into the function gives\( h(x) = (0)^2 - 4 = -4 \). Therefore, the vertex is at \( (0, -4) \).
03

Sketch the Axis of Symmetry

The axis of symmetry for the quadratic function \( y = ax^2 + bx + c \) is a vertical line through the x-coordinate of the vertex. Thus, the axis of symmetry is the line \( x = 0 \).
04

Identify Additional Points

To sketch the graph, identify additional points. Choose \( x = 1 \) and \( x = -1 \) as they are equidistant from the axis of symmetry. Substitute into \( h(x) = x^2 - 4 \):\( h(1) = 1^2 - 4 = -3 \) and \( h(-1) = (-1)^2 - 4 = -3 \). Thus, points are \( (1, -3) \) and \( (-1, -3) \).
05

Sketch the Graph

Plot the vertex \( (0, -4) \) and the additional points \( (1, -3) \) and \( (-1, -3) \). Draw a smooth U-shaped curve passing through these points. Label the vertex and the axis of symmetry. The graph opens upwards since \( a > 0 \).
06

Verify Symmetry

Ensure that the graph is symmetric about the vertical line \( x = 0 \) by checking points equidistant from \( x = 0 \), such as \( (1, -3) \) and \( (-1, -3) \).

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Key Concepts

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

Vertex
In a quadratic function, the vertex is a crucial point that defines the graph's peak or trough. For the function \( h(x) = x^2 - 4 \), the vertex can be found using the formula \[ x = -\frac{b}{2a} \]
  • Here, \( a = 1 \) and \( b = 0 \), so \( x = 0 \).
  • Substitute \( x = 0 \) into the function to find the y-coordinate: \( h(x) = (0)^2 - 4 = -4 \).
Thus, the vertex of the function is at the point \( (0, -4) \). In this context, the vertex is the lowest point on the graph since the parabola opens upwards. Understanding the vertex is essential as it represents the function's minimum or maximum value.
Axis of Symmetry
The axis of symmetry in a quadratic function is a vertical line that passes through the vertex. It divides the parabola into two mirror-image halves. For the function \( h(x) = x^2 - 4 \), the axis of symmetry is determined by the x-coordinate of the vertex.
  • Since the vertex is at \( (0, -4) \), the axis of symmetry is the line \( x = 0 \).
Visualizing the axis of symmetry helps analyze the graph's balance. It ensures that equal values of x on either side have the same y-value, maintaining the parabola's symmetric nature. Recognizing this line makes sketching and understanding quadratic graphs a lot more straightforward.
Graph Sketching
Graph sketching involves creating a visual representation of a quadratic function, making it easier to interpret. To sketch \( h(x) = x^2 - 4 \):
  • Start by plotting the vertex \( (0, -4) \).
  • The axis of symmetry, \( x = 0 \), divides the graph into two equal parts.
  • Pick additional points such as \( (1, -3) \) and \( (-1, -3) \), which are equidistant from the axis of symmetry.
Connect these points with a smooth, continuous U-shaped curve. Since \( a = 1 \) is positive, the parabola opens upwards. Remember to verify symmetry by checking that points on opposite sides of the axis have the same y-value. Graph sketching provides a visual way to summarize the behavior of the quadratic function.

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Most popular questions from this chapter

Use the formula \(A=P(1+r)^{t}\) to solve Exercises 75 through 78 . See Example \(9 .\) Find the rate \(r\) at which \(\$ 3000\) compounded annually grows to \(\$ 4320\) in 2 years.

Find the vertex of the graph of each quadratic function. Determine whether the graph opens upward or downward, find any intercepts, and sketch the graph. See Examples 1 through 4 . $$ f(x)=x^{2}-4 x+5 $$

Without solving, determine whether the solutions of each equation are real numbers or complex but not real numbers. See the Concept Check in this section. $$ (2 y-5)^{2}+7=3 $$

Find the vertex of the graph of each quadratic function. Determine whether the graph opens upward or downward, find the \(y\) -intercept, approximate the \(x\) -intercepts to one decimal place, and sketch the graph. $$ f(x)=3 x^{2}-6 x+7 $$

Use the formula \(A=P(1+r)^{t}\) to solve Exercises 75 through 78 . See Example \(9 .\) Find the rate at which \(\$ 15,000\) compounded annually grows to \(\$ 16,224\) in 2 years.
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