Problem 80 - A Graphical Approach to Precalculus with Limits

Question

Use the analyric 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 \(a\) calculator and the standand 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 for Y-Axis Symmetry

To determine symmetry with respect to the y-axis, check if replacing \(x\) with \(-x\) results in the original function. If \(f(-x) = f(x)\), the function is symmetric about the y-axis. Given \(f(x) = 0.75x^2 + |x| + 1\), first replace \(x\) with \(-x\): \[f(-x) = 0.75(-x)^2 + |-x| + 1\] Simplify: \[f(-x) = 0.75x^2 + |x| + 1\] Since \(f(-x) = f(x)\), the function is symmetric with respect to the y-axis.

Step 2: Check for Origin Symmetry

For origin symmetry, replace \(x\) with \(-x\) and \(f(x)\) with \(-f(x)\). If \(f(-x) = -f(x)\), then the function is symmetric about the origin.We already have \(f(-x) = 0.75x^2 + |x| + 1\) from Step 1.Substitute into \(-f(x)\):\[-f(x) = -(0.75x^2 + |x| + 1) = -0.75x^2 - |x| - 1\]Since \(f(-x) eq -f(x)\), the function is not symmetric about the origin.

Step 3: Graphical Verification

Use a calculator or graphing tool to plot \(f(x) = 0.75x^2 + |x| + 1\) in the standard window setting. Observe the graph.The graph should visually confirm symmetry about the y-axis as both sides (left and right of the y-axis) appear identical, reinforcing the analytical conclusion from Step 1. The graph does not exhibit symmetry about the origin.

Key Concepts

Y-Axis Symmetry

The concept of y-axis symmetry plays a crucial role in analyzing graphs of functions. A function is said to be symmetric with respect to the y-axis if the left and right sides of the graph mirror each other when divided by the y-axis. This means that for any x-value, its negative counterpart should yield the same function value.

To test for y-axis symmetry analytically, we replace every occurrence of x in the function with -x. If the function remains unchanged, i.e., if \(f(-x) = f(x)\), then the function is symmetric about the y-axis.

In our problem, starting with \(f(x) = 0.75x^2 + |x| + 1\), we substitute \(-x\) for \(x\):

• First, replace \(x\) with \(-x\) yielding the expression: \(f(-x) = 0.75(-x)^2 + |-x| + 1\).
• On simplifying, we get \(f(-x) = 0.75x^2 + |x| + 1\) which is equal to \(f(x)\).

Thus, the function is symmetric about the y-axis, as the change did not alter the original expression.

Origin Symmetry

Origin symmetry is another fascinating aspect of function graphs. A function is symmetric with respect to the origin if rotating the graph 180 degrees around the origin results in the same graph. This implies for every point \((x, y)\), there should be a point \((-x, -y)\) as well.

To determine if a graph is symmetric about the origin, replace x with -x and y (or \(f(x)\)) with -y (or \(-f(x)\)). If \(f(-x) = -f(x)\), then the function is origin symmetric.

In our instance, starting with \(f(x) = 0.75x^2 + |x| + 1\), we substitute \(-x\) and check whether it equals \(-f(x)\):

• We have already calculated \(f(-x) = 0.75x^2 + |x| + 1\).
• Now, for \(-f(x)\), we derive \(-f(x) = -(0.75x^2 + |x| + 1) = -0.75x^2 - |x| - 1\).

Since \(f(-x)\) does not equal \(-f(x)\), we conclude that the function does not possess origin symmetry.

Graphical Verification

Graphical verification is a crucial step in confirming analytical deductions about symmetry. While algebraic methods provide a solid foundation, visual verification can solidify understanding and offer a more tangible perspective.

To verify symmetry graphically, we utilize a graphing calculator or graphing software to plot the given function. In the context of our exercise, plot the function \(f(x) = 0.75x^2 + |x| + 1\) using standard window settings.

While observing the graph:

• You'll notice the graph mirrors itself along the y-axis, consistent with our earlier analytical finding of y-axis symmetry.
• However, there is no symmetry evident around the origin, aligning with the analytical conclusion.

Graphical verification serves as a visual confirmation of theoretical analysis and is an essential part of learning to interpret function graphs effectively.