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Graphing exponential functions requires a solid understanding of both the base form of the function and any transformations applied to it.
First, it is important to identify the base exponential function. In the exercise provided, the base function is \( h(x) = 5^x \).
Exponential functions like this have certain characteristics. As \( x \) increases, the \( y \) values grow rapidly if the base is greater than one, resulting in a steep upward curve. Conversely, as \( x \) becomes more negative, the function approaches zero, but never actually touches zero, creating a horizontal asymptote along the x-axis.
When graphing, a useful step is to calculate key points for several values of \( x \). For example, for \( x = -1, 0, 1 \), corresponding \( y \) values would be \( 0.2, 1, \) and \( 5 \) respectively. Plotting these helps visualize how the graph behaves.
Key Points for Graphing:

• Determine the base function and its base "5" plot.
• Identify the direction of the curve as \( x \) increases or decreases.
• Note the horizontal asymptote at \( y = 0 \).

Each of these points ensures you have a solid framework from which to begin your sketch.

Transformations in exponential functions involve changes to the function that result in shifts, stretches, or compressions of the graph. The most straightforward transformation involves shifting the graph horizontally, vertically, or both, which is determined by constants added to the variable or function.
For \( h(x) = 5^{x-2} \), consider the transformation from the original function \( 5^x \). This adjustment suggests a shift rather than a compression or stretch.
When the form is \( 5^{x - c} \), it represents a horizontal shift due to the constant \(-c\) inside the exponent. Consequently, the graph moves without altering the function's shape.
Transformations Breakdown:

• A transformation inside the exponent relates to horizontal shifts.
• Transformations outside the base can affect the vertical shifts, stretches, or compressions.
• Keep the original base curve shape steady to apply other transformations correctly.

Thus, in this scenario, we see there's simply a horizontal shift. Understanding these transformation mechanisms is crucial for accurate graph interpretation.

Horizontal shifts in exponential functions involve moving the entire graph left or right across the x-axis. For \( h(x) = 5^{x-2} \), the graph moves horizontally due to the constant \(-2\) within the exponent.
This negative sign indicates a shift to the right, opposite to what might initially be expected because \( x \) is effectively increased by 2 before evaluating the exponential term.
Thus, a point \((x, y)\) on the base function \(5^x\) becomes \((x+2, y)\) for the transformed function.
Predicting the shift ahead of time helps in ironing out where new critical points will lie on your graph. This directly influences where the rise of the graph displays its signature steep growth.
Steps for Horizontal Shifts:

• Assess the sign and value of the constant in the exponent for direction and magnitude.
• Shift all base function points accordingly to plot the new curve.
• Keep track of asymptotes, which remain unaffected by horizontal shifts.

This consistent procedure ensures efficiency in plotting a robust graph of any transformed exponential function.