Problem 68 - A Graphical Approach to College Algebra

Question

Use the analytic method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use your calculator and the standard window to support your conclusion. $$f(x)=0.75 x^{2}+|x|+1$$

Step-by-Step Solution

Answer

The function is symmetric with respect to the y-axis.

Step 1: Check Symmetry with Respect to the y-axis

For a function to be symmetric with respect to the y-axis, the condition \( f(-x) = f(x) \) must hold true for all \( x \). Let's test this condition for the given function: \[ f(-x) = 0.75(-x)^2 + |-x| + 1 = 0.75x^2 + |x| + 1 \]Since \( f(-x) = f(x) \), the function is symmetric with respect to the y-axis.

Step 2: Check Symmetry with Respect to the Origin

For the graph of a function to be symmetric with respect to the origin, the condition \( f(-x) = -f(x) \) must hold for all \( x \). Let's verify this:\[ f(-x) = 0.75(-x)^2 + |-x| + 1 = 0.75x^2 + |x| + 1 \]\(-f(x) = -(0.75x^2 + |x| + 1) = -0.75x^2 - |x| - 1 \)Since \( f(-x) eq -f(x) \), the function is not symmetric with respect to the origin.

Step 3: Calculator Verification

Use a graphing calculator to plot the function \( f(x) = 0.75x^2 + |x| + 1 \) in the standard window settings. Observe the symmetry of the graph visually. The graph should appear mirrored across the y-axis, which supports our analytical conclusion.

Key Concepts

Y-Axis Symmetry

Y-axis symmetry in a graph means that if you fold the graph over the y-axis, both halves will match perfectly. This is an essential concept because it helps you understand the function's behavior and its relation to geometry. To check for y-axis symmetry in a function analytically, you replace every occurrence of 'x' in the function with '-x.' Then, compare the resulting expression to the original function.

For our function,

• Starting with: \( f(x) = 0.75x^2 + |x| + 1 \)
• Substitute -x: \( f(-x) = 0.75(-x)^2 + |-x| + 1 \)
• Since \( |-x| \) is the same as \( |x| \), the function simplifies to \( f(-x) = 0.75x^2 + |x| + 1 \)
• This gives us \( f(-x) = f(x) \)

This demonstrates that the graph of the function is symmetric concerning the y-axis.

This property of symmetry helps in sketching graphs and understanding the broader behavior of quadratic and absolute value expressions.

Origin Symmetry

Origin symmetry refers to a condition where rotating a graph 180 degrees around the origin leaves it unchanged. This type of symmetry is verified when replacing 'x' with '-x' in the function yields an expression identical to the negative of the original function, \( f(-x) = -f(x) \).

However, for our given function, the following happens:

• We already calculated \( f(-x) \) to be \( 0.75x^2 + |x| + 1 \).
• The negative of the function, \(-f(x)\), is \(-0.75x^2 - |x| - 1 \).
• Since \( f(-x) eq -f(x) \), our function does not possess origin symmetry.

Understanding origin symmetry emphasizes how a function behaves about the coordinate plane. This provides crucial insight into the nature of solutions when solving algebraic expressions, and helps in visualizing complex transformations.

Graphing Calculators

Graphing calculators are powerful tools in checking the symmetry of functions. They visually plot equations, providing an effective way to confirm analytical results. Here's how to use a graphing calculator to verify our function's symmetry:

1. **Inputting the Function**: Enter \( f(x) = 0.75x^2 + |x| + 1 \) into the calculator and set it to the standard window settings.

2. **Observing the Graph**: Notice how the graph appears symmetric about the y-axis. The mirrored shape along the y-axis aligns with our previous analytical findings.

3. **Confirming Analytical Work**: The calculator helps us visually confirm that the function lacks origin symmetry. The 180-degree rotation check doesn't apply, indicating that the shape isn't replicated across all quadrants.

Using graphing calculators not only strengthens understanding of mathematical concepts but also assists in accurate problem solving by validating theoretical methods in a straightforward manner.