Problem 53 - Intermediate Algebra

Question

Is the graph of \(f(x)=x^{2}+2 x+4\) a \(y\)-axis reflection of \(f(x)=x^{2}-2 x+4\) ? Defend your answer.

Step-by-Step Solution

Answer

Yes, the graph is a y-axis reflection.

Step 1: Understanding the Graph Transformation

To determine if one function is a reflection of the other, we need to know how the transformations affect the graph. A reflection about the y-axis changes the sign of the x-coordinates, giving the formula for reflection of a function as: \[ f(-x) = x^2 - 2x + 4 \].

Step 2: Compare Functions

Evaluate the negative input for the first function: \( f(-x) = (-x)^2 + 2(-x) + 4 \). Simplifying this gives: \[ f(-x) = x^2 - 2x + 4 \].

Step 3: Verify the Reflection

Now compare the result with the second function given: \( f(x) = x^2 - 2x + 4 \). Since \( f(-x) \) matches exactly with the second function, the graph of \( f(x) = x^2 + 2x + 4 \) is indeed a y-axis reflection of \( f(x) = x^2 - 2x + 4 \).

Key Concepts

Exploring Quadratic Functions

Quadratic functions are a fundamental type of polynomial function, characterized by the highest power of the variable being two. They are usually expressed in the standard form as: \[ f(x) = ax^2 + bx + c \]where \(a\), \(b\), and \(c\) are constants. The graph of a quadratic function is a parabola which can open upwards or downwards depending on the sign of \(a\). If \(a > 0\), the parabola opens upwards resembling a U-shape, whereas if \(a < 0\), it opens downwards, looking like an inverted U.
Basic characteristics of quadratic functions include:

• Vertex: The highest or lowest point on the graph, which can be found using the formula \(-\frac{b}{2a}\) to determine the x-coordinate.
• Axis of symmetry: A vertical line through the vertex that divides the parabola into two mirror images. The equation for this line is \(x = -\frac{b}{2a}\).
• Roots or Zeros: Points where the parabola crosses the x-axis, found by solving \(ax^2 + bx + c = 0\).

Each quadratic function has its own unique set of these attributes, but their basic form ensures the graphs possess uniformly parabolic shapes.

Understanding Y-axis Reflection

Y-axis reflection is a type of graph transformation where the graph of a function is flipped over the y-axis. This transformation alters the x-coordinates of every point on the graph, effectively changing their sign. The formula for reflecting a function about the y-axis is given by substituting \(x\) with \(-x\) in the function, resulting in \(f(-x)\).
Consider the function \(f(x) = x^2 + 2x + 4\). To determine its y-axis reflection, we substitute \(x\) with \(-x\): \[f(-x) = (-x)^2 + 2(-x) + 4 \]This simplifies to: \[f(-x) = x^2 - 2x + 4 \]Notice how the equation transforms, demonstrating a perfect reflection. This new function, \(f(-x)\), represents the original parabola flipped across the y-axis.
Graphically, each point \((x, y)\) on the original function is mirrored to \((-x, y)\) on the reflected function. This type of reflection does not affect the y-coordinates of the graph points.

Mastering Function Evaluation

Function evaluation is a crucial skill in understanding and working with mathematical functions. It involves substituting a particular value of \(x\) into the function to find the corresponding output or y-coordinate. Evaluating functions helps to graph the function by providing specific points that can be plotted on a coordinate plane.
For the function \(f(x) = x^2 + 2x + 4\), to evaluate \(f\) for a given value of \(x\), you substitute that value into the function. For instance, if you want to evaluate the function at \(x = 1\):

• Replace \(x\) with 1: \(f(1) = 1^2 + 2(1) + 4\)
• Calculate the result: \(f(1) = 1 + 2 + 4 = 7\)

This process works similarly for any value you choose to substitute into the function. It's particularly useful when checking the outcomes of transformations like reflections, ensuring that the output from reflected functions behaves as expected.