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Chapter 2: Problem 85

These exercises show how the graph of \(y=|f(x)|\) is obtained from the graph of \(y=f(x)\). The graphs of \(f(x)=x^{2}-4\) and \(g(x)=\left|x^{2}-4\right|\) are shown. Explain how the graph of \(g\) is obtained from the graph of \(f\). (GRAPH CANT COPY) $$f(x)=x^{2}-4 \quad g(x)=\left|x^{2}-4\right|$$

Short Answer

Expert verified
Reflect portions of the graph of \(f(x) = x^2 - 4\) below the x-axis above it to obtain \(g(x) = |x^2 - 4|\).

Step by step solution

01

Identify the Function

Start with the function given, which is \(f(x) = x^2 - 4\). This is a quadratic function, a parabola that opens upwards, with its vertex at \((0, -4)\). It intersects the x-axis at the points where \(x^2 = 4\), namely \(x = -2\) and \(x = 2\).
02

Understand the Absolute Value Transformation

The graph of \(g(x) = |x^2 - 4|\) involves taking the absolute value of \(f(x)\). The absolute value operator \(|\cdot|\) reflects all parts of the graph of \(f(x)\) that are below the x-axis (negative values of \(f(x)\)) to be above the x-axis (become positive).
03

Draw the Original Graph

Plot the graph of \(f(x) = x^2 - 4\). This parabola crosses the x-axis at \((-2,0)\) and \((2,0)\), and has its vertex at \((0, -4)\). It is symmetrical about the y-axis.
04

Apply the Absolute Value

For the graph of \(y = |f(x)|\), wherever \(f(x)\) is negative (below the x-axis), reflect this portion above the x-axis. This means that the section from \(-2 < x < 2\) which was below the x-axis in the \(f(x)\) graph will be flipped upwards in the \(g(x)\) graph. Outside the interval \(-2, 2\), the graph remains unchanged because \(f(x)\) is already positive or zero.
05

Graph the Transformed Function

After the transformation, the graph of \(g(x)\) retains the shape of the parabola for \(|x| \geq 2\), but dips at the line \(x = 0\) from the x-axis at points \((-2,0)\) and \((2,0)\) like a 'V' shape within \(-2 < x < 2\). Thus, the graph is symmetric and the part originally below the x-axis in \(f(x)\) now mirrors above it.

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Key Concepts

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

Quadratic Functions
Quadratic functions are at the heart of many algebraic concepts. The standard form of a quadratic function is given by \( f(x) = ax^2 + bx + c \). In this equation, a, b, and c are constants, and \( a eq 0 \). Such functions are known for creating a U-shaped curve on a graph which is called a parabola.

Here are some key characteristics of quadratic functions:
  • **Parabola shape**: The graph of a quadratic function is called a parabola. If \( a > 0 \), the parabola opens upwards, resembling a U shape. If \( a < 0 \), it opens downwards, forming a n shape.
  • **Vertex**: The highest or lowest point of the parabola is called the vertex. For the function \( f(x) = x^2 - 4 \), the vertex is at the point \((0, -4)\).
  • **Axis of symmetry**: This is a vertical line that runs through the vertex, splitting the parabola into two mirror-image halves. For \( f(x) = x^2 - 4 \), the axis of symmetry is \( x = 0 \).
  • **Roots or zeroes**: These are the x-values where the quadratic touches or crosses the x-axis. In our example, the roots occur at \( x = -2 \) and \( x = 2 \).
Understanding these elements is essential for graphing quadratic functions and predicting their behavior.
Parabola Transformation
Transformations refer to shifting, reflecting, stretching, or compressing a graph. For a quadratic function like \( f(x) = x^2 - 4 \), transformations can help us understand how the function changes behavior. One common transformation is translating or moving a function up, down, left, or right.

Here's how transformations relate to quadratic functions:
  • **Vertical shifts**: Moving the entire graph up or down without changing its shape. This happens when we add or subtract a number to the function.
  • **Horizontal shifts**: Moving the graph left or right. This can occur when every x-value in the function is adjusted by a certain number.
  • **Reflections**: Flipping the graph over a line, such as the x-axis or y-axis, which we'll discuss more in the next section.
  • **Stretching and compressing**: Changing the width of the parabola by multiplying \( x \) by a constant. These adjustments make the parabola look narrower or wider.
The graph of the transformed function \( g(x) = |x^2 - 4| \) utilizes reflection across the x-axis due to the absolute value function, which is a distinct type of transformation.
Graph Reflection
When you see a transformation involving absolute values, you often encounter reflection. Any point on a graph that's reflected will maintain the same distance from the line over which it鈥檚 flipped. For our quadratic function, reflection plays a vital role when the absolute value function is introduced.

Here's how reflection works in this context:
  • **Using absolute value**: Applying the absolute value, like in \( g(x) = |x^2-4| \), reflects any part of the graph of \( f(x) \) that is below the x-axis to above. Negative y-values become positive.
  • **Impact on the graph**: In the function \( f(x) = x^2 - 4 \), values between \( -2 < x < 2 \) create negative results. The absolute value transformation converts these values into a mirrored positive V-shape.
  • **Symmetry**: The original symmetry of the parabola remains but with this flip involved. The graph appears like a 'V' in the transformed portion while maintaining the upward opening structure beyond \( x = -2 \) and \( x = 2 \).
Understanding reflection will help you grasp how these transformations result in the final graph of \( g(x) = |x^2 - 4| \).

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