Problem 50 - Algebra and Trigonometry

Question

45–50 ? Test the equation for symmetry. $$ y=x^{2}+|x| $$

Step-by-Step Solution

Answer

The graph \( y = x^2 + |x| \) is symmetric about the y-axis only.

Step 1: Understand the Equation

The equation given is \( y = x^2 + |x| \). It is composed of two parts: the quadratic \( x^2 \) and the absolute value \( |x| \). We need to test this equation for symmetry with respect to the y-axis, x-axis, and origin.

Step 2: Test Symmetry with Respect to the Y-axis

To test for symmetry with respect to the y-axis, substitute \(-x\) for \(x\) in the equation: \[ y = (-x)^2 + |-x| \].Simplifying, we have \( y = x^2 + |x| \). Since the equation remains the same, this indicates the graph is symmetric about the y-axis.

Step 3: Test Symmetry with Respect to the X-axis

Replace \(y\) with \(-y\) in the equation: \[ -y = x^2 + |x| \].To test if this equation is equivalent to the original, rearrange it for \(y\): \[ y = -x^2 - |x| \]. The resulting equation is not equivalent to the original; hence, the graph is not symmetric about the x-axis.

Step 4: Test Symmetry with Respect to the Origin

Replace \(x\) with \(-x\) and \(y\) with \(-y\) in the original equation: \[ -y = (-x)^2 + |-x| \].This simplifies to \( -y = x^2 + |x| \), which is not equivalent to \( y = x^2 + |x| \). Therefore, the graph is not symmetric about the origin.

Key Concepts

Quadratic Functions

When exploring quadratic functions, you're looking at equations that take the form \( y = ax^2 + bx + c \). This particular shape is called a parabola. The graph of a basic quadratic function, \( y = x^2 \), is a U-shaped curve that opens upward. Quadratic functions are important because they capture the behavior of many natural phenomena, like the path of a thrown ball or the spread of light.

• **Vertex**: The highest or lowest point of a parabola.
• **Axis of Symmetry**: A vertical line that passes through the vertex, dividing the parabola into two mirror-image halves.
• **Direction**: If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.

The function \( y = x^2 + |x| \) includes both **quadratic** and **absolute value** components. This combination affects the symmetry of the graph as seen in our original problem. The quadratic part \( x^2 \) contributes to symmetry about the **y-axis**, forming a strong foundation for testing other symmetries.

Absolute Value

Absolute value represents the distance of a number from zero on the number line, without considering direction. The expression \( |x| \) will always yield a non-negative result, which changes how equations behave graphically.
For example, \( y = |x| \) produces a V-shaped graph that opens upwards, with the point of the V located at \( x=0 \). This part of the equation also maintains symmetry around the **y-axis**.
When combined with the quadratic part \( x^2 \), absolute value helps define the equation's overall symmetry.

• Graphs involving absolute value tend to have sharp turns, like a corner at the vertex of \( y = |x| \).
• The blend of \( x^2 \) and \(|x|\) in the original problem reinforces **y-axis symmetry** due to these even functions.

Thus, studying absolute value functions provides insights into how they alter graph behaviors and symmetry tests.

Axis of Symmetry

The axis of symmetry is a crucial aspect of understanding graph symmetry in mathematics, especially regarding quadratic and absolute value functions. Simply put, it's a line through a graph that the graph could be "folded" along to create matching halves.
For quadratic functions, the axis of symmetry is given by the formula \( x = -\frac{b}{2a} \), derived from the standard quadratic form \( y = ax^2 + bx + c \).
In our function case, both \( x^2 \) and \( |x| \) suggest symmetry about the **y-axis** because:

• The quadratic term, \( x^2 \), is symmetric around this axis inherently because squaring a number makes it positive.
• The absolute value term, \( |x| \), is similarly symmetric due to returning positive values regardless of input sign.

So, the y-axis stands as the axis of symmetry for this specific equation. Learning about the axis of symmetry helps identify symmetry properties quickly, thus aiding the testing process for any function's graph.