Problem 74 - Algebra and Trigonometry Real Mathematics, Real People

Question

Check for symmetry with respect to both axes and to the origin. Then determine whether the function is even, odd, or neither. $$(x-2)^{2}=y+4$$

Step-by-Step Solution

Answer

The function \((x-2)^{2}=y+4\) shows no symmetry with respect to either axes or the origin. The function is neither even nor odd.

Step 1: Transform the given equation

First we rewrite the given equation in the form of \(y = f(x)\): \((x-2)^{2}=y+4\) becomes \(y = (x-2)^2 - 4\).

Step 2: Test for symmetry with respect to x-axis

Replace \(y\) with \(-y\) and check if we get the same equation. This method changes the equation into \(-y = (x-2)^2 - 4\) or \(y = -((x-2)^2 - 4)\), which is not equal to our original equation so it's not symmetric with respect to the x-axis.

Step 3: Test for symmetry with respect to y-axis

Replace \(x\) with \(-x\) in the transformed equation. The equation changes into \(y = ((-x)-2)^2 - 4 = (x+2)^2 - 4\). As it is different from the original equation, this shows that it is not symmetric with respect to the y-axis.

Step 4: Test for symmetry with respect to the origin

Replace both \(x\) and \(y\) with \(-x\) and \(-y\) respectively. The equation changes into \(-y = ((-x)-2)^2 - 4 = (x+2)^2 - 4\), which is not equal to the original equation. Hence, it is not symmetric with respect to the origin.

Step 5: Determine whether the function is even, odd, or neither

As we have already implemented the transformations in steps 3 and 4, it is clear that the function is not even \(f(x) != f(-x)\) and it is also not odd \(f(-x) != -f(x)\), so the function is neither even nor odd.

Key Concepts

Function Symmetry

When exploring function symmetry, we're looking at whether a graph reflects or aligns in certain ways. There are three main types:

• Symmetry with respect to the x-axis: If replacing \(y\) with \(-y\) yields the same equation, the graph is symmetric about the x-axis. For many functions, such as quadratic ones like \(y = (x-2)^2 - 4\), this symmetry is not present.


• Symmetry with respect to the y-axis: If replacing \(x\) with \(-x\) results in the same equation, it's symmetric about the y-axis. Simple polynomials like \(x^2\) exhibit this symmetry, but our function's modification \((x+2)^2\) shows it’s not y-axis symmetric.


• Symmetry with respect to the origin: This occurs if substituting both \(x\) with \(-x\) and \(y\) with \(-y\) keeps the equation unchanged. Origin symmetry typically occurs in odd functions, which is not the case here.

The exercise examines these symmetries and finds none are present in \(y = (x-2)^2 - 4\).

Even and Odd Functions

Identifying whether a function is even or odd is crucial for understanding its behavior. Here’s how you can determine this:

• Even Functions: A function is even if \(f(x) = f(-x)\). Graphically, this means the function is symmetric about the y-axis. Commonly, functions like \(x^2\) are even, but the transformation in our function \((x-2)^2\) means it's not.


• Odd Functions: A function is odd if \(f(-x) = -f(x)\). This introduces symmetry about the origin. Functions like \(x^3\) fit this category.

In our case, \(f(x) eq f(-x)\) and \(f(-x) eq -f(x)\), concluding that the function \((x-2)^2 - 4\) is neither even nor odd.

Quadratic Equations

Quadratic equations are fundamental in algebra and appear in the form \(ax^2 + bx + c\). Our given function \((x-2)^2 - 4\) exemplifies this where it's easier to view as:\[\begin{align*}(x-2)^2 &= y + 4 \y &= (x-2)^2 - 4 \end{align*}\]Here, the equation's structure gives insight into the graph being a parabola.

**Key Properties of Quadratics:**

• Vertex Form: Our function is already in vertex form \( (x-h)^2 + k \), where \( h=2 \) and \( k=-4 \), making \((2, -4)\) the vertex.


• Axis of Symmetry: The axis of symmetry is always \(x=h\) for the vertex form, which is \(x=2\) in this case.


• Direction of Opening: Since the term \((x-2)^2\) is positive, the parabola opens upwards.

Understanding these properties is key in mastering how quadratic equations behave graphically and algebraically.