Question

Match each function with the corresponding transformation of \(y=\left(\frac{3}{5}\right)^{x}\). a) \(y=\left(\frac{3}{5}\right)^{x+1}\) b) \(y=-\left(\frac{3}{5}\right)^{x}\) c) \(y=\left(\frac{3}{5}\right)^{-x}\) d) \(y=\left(\frac{3}{5}\right)^{x}-2\) \(\mathbf{A}\) reflection in the \(x\) -axis \(\mathbf{B}\) reflection in the \(y\) -axis \(\mathbf{C}\) translation down \(\mathbf{D}\) translation left

Short answer

a) D, b) A, c) B, d) C

Step-by-step solution

Identify Base Function

The base function given is \( y = \left(\frac{3}{5}\right)^{x} \). This will be used for comparison with other functions.

Analyze Transformation for y = \left(\frac{3}{5}\right)^{x+1}

Adding 1 to the exponent (\(x+1\)) shifts the graph horizontally. Specifically, it results in a leftward translation because the function \( y = \left(\frac{3}{5}\right)^{x + k} \) moves left if \( k > 0 \). Thus, this corresponds to a translation left.

Analyze Transformation for y = -\left(\frac{3}{5}\right)^{x}

The negative sign in front of the function indicates a reflection over the x-axis. The graph of \( y = - f(x) \) is a reflection of \( y = f(x) \) over the x-axis.

Analyze Transformation for y = \left(\frac{3}{5}\right)^{-x}

Changing the sign of the exponent (to \(-x\)) flips the graph horizontally, indicating a reflection over the y-axis. The function \( y = f(-x) \) reflects \( y = f(x) \) over the y-axis.

Analyze Transformation for y = \left(\frac{3}{5}\right)^{x}-2

Subtracting 2 from the function translates the graph downward by 2 units. The function \( y = f(x) - k \) translates the graph downward if \( k > 0 \).

Key concepts

horizontal translation

When you see a transformation like \( y = \left( \frac{3}{5} \right)^{x+1} \), the crucial detail is the addition inside the exponent. Horizontal translations involve shifts left or right along the x-axis.
In this case, adding 1 to the exponent of x (as in \( x + 1 \)) means moving the graph to the left by 1 unit.
Remember, shifts to the left occur when you add a positive number to x within the function:

• If you see \( y = f(x+a) \), the graph shifts left by 'a' units
• If it's \( y = f(x-a) \), it goes right by 'a' units.

Horizontal shifts don't affect the shape or orientation of the graph, just its position along the x-axis.

vertical translation

Next, consider a transformation like \( y = \left( \frac{3}{5} \right)^{x} - 2 \). This is a vertical translation.
Subtraction of a number from the entire function moves the graph downward by that number of units.
Vertical translations involve shifts up or down along the y-axis:

• \( y = f(x) - a \) means shifting down by 'a' units.
• Conversely, \( y = f(x) + a \) means shifting up by 'a' units.

These shifts also don't distort the graph's shape, only its vertical position.
In this example, subtracting 2 indicates the entire graph moves down by 2 units.

reflection over x-axis

Now, let’s explore reflections, starting with reflections over the x-axis like \( y = - \left( \frac{3}{5} \right)^{x} \).
The negative sign in front of the function reflects the graph across the x-axis.

• This means every point on the original graph is flipped to the opposite side of the x-axis.
• For example, if the original function was above the x-axis, after the transformation, it will be below the x-axis.

Reflections over the x-axis change the values of y but leave the x-coordinates unchanged.

reflection over y-axis

Lastly, let's discuss reflections over the y-axis using \( y = \left( \frac{3}{5} \right)^{-x} \).
The negative sign in the exponent indicates a reflection across the y-axis.
Each point on the graph of the original function is flipped over to the opposite side of the y-axis.
The transformation can be understood by these points:

• Reflections over the y-axis change the x-coordinates of the graph’s points but leave the y-values unchanged.

So, if a point (x, y) was on the original graph, after the reflection, the point becomes (-x, y).