Problem 70 - College Algebra with Modeling and Visualization

Question

For the given representation of a function \(f,\) graph the reflection across the \(x\)-axis and graph the reflection across the \(y\)-axis. \(f(x)=4-7 x-2 x^{2}\)

Step-by-Step Solution

Answer

Graph \\(-f(x) = -4 + 7x + 2x^2\\) for x-axis reflection and \\(f(-x) = 4 + 7x - 2x^2\\) for y-axis reflection.

Step 1: Understanding Reflection Across the x-axis

When a function is reflected across the x-axis, each point \((x, y)\) on the graph of the function is transformed to \((x, -y)\). Practically, this means we take the negative of the function's output. Thus, for the function \(f(x) = 4 - 7x - 2x^2\), the reflection across the x-axis is \(-f(x) = - (4 - 7x - 2x^2) = -4 + 7x + 2x^2\).

Step 2: Graph Reflection Across the x-axis

Graph the function \(-f(x) = -4 + 7x + 2x^2\). Notice the change in the leading coefficient and the constant. This reflects the parabola, flipping it vertically around the x-axis, and changing the direction of its opening.

Step 3: Understanding Reflection Across the y-axis

A reflection across the y-axis means that each point \((x, y)\) on the graph is transformed to \((-x, y)\). This is done by replacing every \(x\) in the function with \(-x\). Hence, for \(f(x) = 4 - 7x - 2x^2\), the reflection across the y-axis is \(f(-x) = 4 + 7x - 2x^2\).

Step 4: Graph Reflection Across the y-axis

Graph the function \(f(-x) = 4 + 7x - 2x^2\). This modifies only the linear term and maintains the nature of the parabola, opening downward, but shifts the direction of the linear component, reflecting horizontally across the y-axis.

Key Concepts

Reflection Across x-axis

Reflecting a function across the x-axis essentially turns the graph upside down. Imagine holding a piece of paper with a drawn graph and flipping it vertically about the horizontal line (x-axis). Here is how it works:

• For every point on the graph, you transform \( (x, y) \) into \( (x, -y) \).
• Mathematically, you take the negative of the whole function. For instance, if your function is \( f(x) = 4 - 7x - 2x^2 \), the reflection will be \(-f(x) = -4 + 7x + 2x^2\).
• This operation flips the entire graph over the x-axis, reversing the direction in which the parabola opens.

The leading coefficient being positive after reflection indicates the parabola opens upward. It is a neat transformation that visually represents taking the opposite of each y-value on the graph. Once you've done this, graphing the new function shows this mirrored shape beautifully.

Reflection Across y-axis

Reflection across the y-axis involves flipping the graph horizontally, as though you’ve turned the page over as a mirror reflection across the vertical line (y-axis). Here's how it's done:

• The transformation changes each point from \( (x, y) \) to \( (-x, y) \).
• In practice, you replace every \( x \) in the original function with \( -x \). For the function \( f(x) = 4 - 7x - 2x^2 \), this becomes \( f(-x) = 4 + 7x - 2x^2 \).
• This changes the linear component, reversing its direction while keeping the parabola's general shape intact.

The parabola still opens downward since the leading coefficient remains negative. However, the graph's left-right orientation is altered, providing a new perspective on the function's behavior across the y-axis.

Graphing Parabolas

Graphing parabolas is an important skill when dealing with quadratic functions. Understanding their features helps in visualizing transformations like reflections. Here’s a simple guide to graphing parabolas:

• The standard form of a quadratic function is \( ax^2 + bx + c \).
• The coefficient \( a \) determines the direction of the parabola’s opening. If \( a \) is positive, the parabola opens upward; if negative, it opens downward.
• The axis of symmetry can be found using \( x = -\frac{b}{2a} \).
• The vertex, where the parabola changes direction, is located at \( (\frac{-b}{2a}, f(\frac{-b}{2a})) \).
• Plotting the vertex and additional points on either side helps in sketching the curve correctly.

With parabolas, shifts, reflections, and stretches are easily spotted on a graph. Parabolas have a symmetrical nature which makes finding points of reflection intuitive. Watching how a parabola changes shape after each transformation ties together the concept of graphing with transformations like reflections.