Problem 67 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function and determine whether it is even, odd, or neither. $$h(x)=x^{2}+6$$

Step-by-Step Solution

Answer

The given function \(h(x)=x^{2}+6\) is an even function because it satisfies the property \(h(-x)=h(x)\) and its graph presents symmetry with respect to the y-axis.

Step 1: Identify the function

The given function is \(h(x)=x^{2}+6\).

Step 2: Check if the function is even

To validate this property, verify if \(h(-x)=h(x)\). If we replace \(x\) with \(-x\) in the function, we get the equation \(h(-x)=(-x)^{2}+6=x^{2}+6\). Therefore, it indeed matches \(h(x)\). This confirms \(h(x)\) is an even function.

Step 3: Use of graphing utility

By using a graphing utility, we can graph the function \(h(x)=x^{2}+6\). The graph will verify the y-axis symmetry, confirming that it is an even function.

Key Concepts

Graphing Utility

Graphing utilities are tools that help visualize mathematical functions. These can be software or calculators capable of graphing equations. For the function \( h(x) = x^2 + 6 \), a graphing utility can plot the graph to show its shape and key characteristics like symmetry.
By inputting the function into the utility, it will produce a curve on a coordinate plane. You observe the graph of \( h(x) = x^2 + 6 \) is a parabola opening upwards due to the \( x^2 \) term.
Graphing utilities make it easy to check if the graph has any symmetry, such as mirroring over the y-axis, which suggests the function may be even. These tools give precise visual confirmation of your mathematical analysis.

Function Symmetry

Symmetry in functions refers to how the graph behaves relative to different axes.
**Even Functions**: If \( h(-x) = h(x) \) for all \( x \), the function is even, meaning it shows symmetry about the y-axis. You will see that the left side of the graph is a mirror image of the right side.
**Odd Functions**: If \( h(-x) = -h(x) \), the function is odd, exhibiting symmetry about the origin. The graph would appear symmetric about the point (0,0).
For the function \( h(x) = x^2 + 6 \), since \( h(-x) \) equals \( h(x) \), the function is even, and its parabola graph confirms this with y-axis symmetry.

Quadratic Function

A quadratic function is any function of the form \( ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants. In our example, \( h(x) = x^2 + 6 \), the function has:
- **Leading Coefficient**: \( a = 1 \) for the \( x^2 \) term, making the parabola open upwards.
- **Constant Term**: \( c = 6 \), which raises the parabola 6 units above the x-axis.
Quadratic functions always graph as parabolas. Depending on the sign of \( a \), they can open up or down. They are fundamental in algebra for solving problems related to areas and maximum/minimum values.

Algebra Concepts

In algebra, understanding the structure of equations is key. Analyzing a function like \( h(x) = x^2 + 6 \) involves:
- **Substituting Values**: For symmetry checking, substitute \(-x\) for \( x \) to compare \( h(-x) \) and \( h(x) \).
- **Recognizing Patterns**: Even powers of \( x \) (like \( x^2 \)) result in even functions. This recognition is vital for predictions before graphing.
- **Simplifying**: Clean up expressions to ensure accurate comparisons between different forms of the equation.
Understanding these concepts allows you to determine function properties quickly and confirm through graphing.