Question

Describe the transformation of \(f\) represented by \(g\). Then graph each function. \(f(x)=e^{-x}, g(x)=e^{-x}+6\)

Short answer

The transformation of function \(f\) to function \(g\) is a vertical shift upward by 6 units. When graphed, both functions have the same overall shape, but \(g(x)\)'s graph is 6 units higher on the y-axis than \(f(x)\)'s graph.

Step-by-step solution

Identify the transformation

Comparing function \(f(x)=e^{-x}\) and \(g(x)=e^{-x}+6\), it can be seen that \(g(x)\) is essentially \(f(x)\) with a +6 added to it. In the context of functions and their graphs, adding a constant to a function shifts the graph of the function vertically. Since we are adding 6, that means our function is shifted 6 units upward. Therefore, \(g(x)\) is a vertical shift of \(f(x)\) 6 units up.

Graph the function \(f(x)\)

Function \(f(x)=e^{-x}\) would be graphed as normally. It's an exponential decay function, starting from a high positive y-value and getting infinitely closer to the x-axis as x increases, and approaching negative infinity as x goes to positive infinity. Note the y-intercept is at \(y=1\).

Graph the function \(g(x)\)

Function \(g(x)=e^{-x}+6\), as determined in Step 1, is simply the graph of \(f(x)\) shifted 6 units up vertically. Therefore, the y-intercept that was previously at \(y=1\) in function \(f(x)\) will now be at \(y=7\). The overall shape of the graph remains identical, just moved up 6 units.

Key concepts

Function Transformation

When we talk about function transformation, we mean changes to the basic function that alter its appearance on a graph. Transformations can include shifting, stretching, compressing, or reflecting the graph. These transformations are applied to functions to better model real-life scenarios or to fit certain constraints or conditions.

In our specific exercise, the transformation is an example of a vertical shift. We take the original function, which is an exponential decay function, and apply a modification that changes its position rather than its shape. Transformations are incredibly useful in mathematics for visualizing and understanding how changes to equations impact the graph.

Vertical Shift

The vertical shift is one of the simplest forms of transformations. It involves moving the entire graph up or down on the coordinate plane without affecting its shape or orientation.

For the functions given in the exercise, the base function is \(f(x)=e^{-x}\). The modified function is \(g(x)=e^{-x}+6\). In this case, the transformation occurs by adding 6 to every output of the original function. Essentially, what happens is:

• Every point on the graph of \(f(x)\) is moved vertically upward by 6 units.
• The y-intercept originally at \(y=1\) now moves to \(y=7\).

The function's shape remains consistent because we only affect the output, not the input or the functional form itself.

Graphing Functions

Graphing functions is a visual way of understanding the relationship between the independent variable (usually \(x\)) and the dependent variable (usually \(y\)). For functions, every input corresponds to exactly one output, making it easy to draw their graphs on a cartesian coordinate plane.

Starting with \(f(x)=e^{-x}\), this is an exponential decay function:

• It decreases as \(x\) increases, getting continually closer to the x-axis.
• The curve starts above the x-axis and never crosses it, instead approaching it asymptotically.
• In the case of exponential decay, the function moves from a y-intercept of 1 and decreases towards zero but never quite reaches it.

For \(g(x) = e^{-x} + 6\):

• The same behavior occurs, except it is shifted upwards by six units across all points due to the vertical shift.
• The new y-intercept becomes 7, maintaining the same exponentially decaying shape relative to the x-axis but higher in absolute position due to the modified equation.

Graphing each function helps us see how transformations like vertical shifts alter only the position, not the fundamental behavior or shape of the function.