Problem 58 - Intermediate Algebra

Question

For each of the following, answer the question on the basis of your knowledge of transformations, and then use a graphing calculator to check your answer. (a) Is the graph of \(f(x)=2 x^{2}+8 x+13\) a \(y\)-axis reflection of \(f(x)=2 x^{2}-8 x+13\) ? (b) Is the graph of \(f(x)=3 x^{2}-12 x+16\) an \(x\)-axis reflection of \(f(x)=-3 x^{2}+12 x-16 ?\) (c) Is the graph of \(f(x)=\sqrt{4-x}\) a \(y\)-axis reflection of \(f(x)=\sqrt{x+4}\) ? (d) Is the graph of \(f(x)=\sqrt{3-x}\) a \(y\)-axis reflection of \(f(x)=\sqrt{x-3}\) ? (e) Is the graph of \(f(x)=-x^{3}+x+1\) a \(y\)-axis reflection of \(f(x)=x^{3}-x+1\) ? (f) Is the graph of \(f(x)=-(x-2)^{3}\) an \(x\)-axis reflection of \(f(x)=(x-2)^{3}\) ? (g) Is the graph of \(f(x)=-x^{3}-x^{2}-x+1\) an \(x\)-axis reflection of \(f(x)=x^{3}+x^{2}+x-1\) ? (h) Is the graph of \(f(x)=\frac{3 x+1}{x}\) a vertical translation of \(f(x)=\frac{1}{x}\) upward 3 units? (i) Is the graph of \(f(x)=2+\frac{1}{x}\) a \(y\)-axis reflection of \(f(x)=\frac{2 x-1}{x} ?\)

Step-by-Step Solution

Answer

(a) No, (b) No, (c) No, (d) No, (e) Yes, (f) Yes, (g) Yes, (h) No, (i) No.

Step 1: Problem Analysis

To determine whether a graph is a reflection over the y-axis or x-axis, we need to compare the functions with respect to even and odd properties. For a y-axis reflection, we check if replacing \(x\) with \(-x\) in one equation results in the corresponding terms of the other equation. An x-axis reflection involves changing the sign of the whole function.

Step 2: Transformation Analysis for Part (a)

For the quadratic functions, compare the coefficients; the term \(2x^2\) remains the same and the constant \(13\) remains unchanged, suggesting no y-axis reflection from \(2x^{2}+8x+13\) to \(2x^{2}-8x+13\) as \(x\) or \(-x\) would produce different results in the linear term.

Step 3: Transformation Analysis for Part (b)

Check if multiplying the entire function by \(-1\) yields the other function. Applying this to \(f(x) = 3x^2 - 12x + 16\) does not give \(-3x^2 + 12x - 16\), because only some terms change signs. Therefore, this is not an x-axis reflection.

Step 4: Transformation Analysis for Part (c)

For the square root function, check if \(x\) is replaced with \(-x\). The function \(\sqrt{4-x}\) is not equivalent to \(\sqrt{x+4}\), as their domains differ, confirming no y-axis reflection occurs.

Step 5: Transformation Analysis for Part (d)

Rewriting \(\sqrt{3-x}\) replacing \(x\) with \(-x\) would not yield the function \(\sqrt{x-3}\). Thus, no y-axis reflection is present.

Step 6: Transformation Analysis for Part (e)

Check if swapping \(x\) with \(-x\) in \(-x^3 + x + 1\) transforms it into \(x^3 - x + 1\). Since each term changes appropriately, this is indeed a y-axis reflection.

Step 7: Transformation Analysis for Part (f)

Verify whether multiplying \((x-2)^3\) by \(-1\) results in \(-((x-2)^3)\). As this operation changes the entire expression's sign, this confirms an x-axis reflection.

Step 8: Transformation Analysis for Part (g)

Check if flipping the sign of the entire function \(x^3 + x^2 + x - 1\) produces \(-x^3 - x^2 - x + 1\). Since the transformation holds, this is indeed an x-axis reflection.

Step 9: Transformation Analysis for Part (h)

Determine if the entire function for vertical translation. Adding \(3\) to \(\frac{1}{x}\) should increase the function by \(3\) units, thus it does not show a match; such translation doesn't apply.

Step 10: Transformation Analysis for Part (i)

Evaluating a y-axis reflection would involve moving \(-x\) resulting in transformantion. There is no uniform reflection forming between \(f(x)=2+\frac{1}{x}\) and \(f(x)=\frac{2x-1}{x}\). This is not a valid y-axis reflection.

Key Concepts

y-axis reflection

A y-axis reflection is a type of transformation that flips the graph of a function about the y-axis. This means each point on the left side of the graph will be mirrored to the right side, and vice versa. To check for a y-axis reflection, replace every instance of the variable \(x\) in the function with \(-x\). If the resulting function matches the original function, it indicates symmetry across the y-axis, thus confirming a y-axis reflection.

In the context of the problems above, consider the example of the polynomial \(f(x) = -x^3 + x + 1\) and \(f(x) = x^3 - x + 1\). By replacing \(x\) with \(-x\) in the first function, \(-x^3\) becomes \(x^3\), and \(x\) becomes \(-x\), which transforms it into the second function. This confirms the y-axis reflection as each term changes predictably:

• \(-x^3\) becomes \(x^3\)
• \(x\) becomes \(-x\)

No transformations occurred to the constant term, as it remains the same.

x-axis reflection

An x-axis reflection involves flipping the entire graph about the x-axis. It is achieved by multiplying the whole function by \(-1\). If negating the entire function results in the counterpart function, the graph reflects over the x-axis. For example, given the exercise function \(f(x) = (x-2)^3\) and its reflection \(f(x) = -(x-2)^3\), notice that multiplying the whole function by \(-1\) converts \((x-2)^3\) to \(-(x-2)^3\).

Here is how the transformation happens in other parts of the problem:

• Flipping the sign of the cubic function \(x^3 + x^2 + x - 1\), yields \(-x^3 - x^2 - x + 1\), matching its pair.

This confirms that an x-axis reflection flips every point of the function across the x-axis, changing positive y-values to negative, and vice versa.

vertical translation

Vertical translation is a transformation that moves the graph of a function up or down without changing its shape. This is achieved by adding or subtracting a constant term from the function. For instance, if you add \(3\) to the function \(f(x) = \frac{1}{x}\), it translates upwards by 3 units, resulting in \(f(x) = \frac{1}{x} + 3\).

Consider the example from the exercise concerning the function \(f(x) = \frac{3x+1}{x}\) and checking if it is a vertical translation of \(f(x) = \frac{1}{x}\). To move a graph vertically by a specific number of units, add the number directly to the function. This means for a translation upward by 3 units, the adjustment should strictly be additions like \(f(x) = \frac{1}{x} + 3\), which the given function does not reflect since it involves a more complex alteration of terms rather than a straightforward vertical movement.

Vertical translation is simple and direct, offering a distinct method to shift the graph in a specific direction along the y-axis.