Question Graph the function $$ f(x)=\left(\frac{1}{4}\right)^{x} $$ by moving the key points. You can only graph integer points, for example you cannot plot the point $$ \left(-1, \frac{1}{4}\right) $$.

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The function f(x) = (1/4)^x is an exponential function. One standard strategy for graphing an exponential is to start with a few “key points.” In our case, we are allowed to plot only points with integer coordinates. Notice that when x is an integer, the output f(x) is not necessarily an integer—but you can still plot the point if you know its exact location. However, if you prefer to have both coordinates as integers, an alternate form is helpful. Step 1. Rewrite the function. Recall that (1/4)^x = 4^(–x). This manipulation shows that if we plug in an integer value for x, then for negative x the y‐values become powers of 4 (which are integers). For example, when x = –2 we get f(–2) = 4^2 = 16. Step 2. Choose several integers for x. Let’s pick a few values for x (both negative, zero, and positive): x f(x) = (1/4)^x or 4^(–x) –2 (1/4)^(–2) = 4^2 = 16 –1 (1/4)^(–1) = 4^1 = 4 0 (1/4)^0 = 1 1 (1/4)^1 = 1/4 (Note: 1/4 is not an integer.) 2 (1/4)^2 = 1/16 (Also not an integer.) Step 3. Plot the “key points” that you are allowed to plot easily. While the instructions say you “cannot plot the point (–1, 1/4)” (which is a misprint since f(–1) = 4, not 1/4) if we were trying to get both coordinates to be integers, we notice that the left-hand side points have integer outputs. So it is best to use: Point A: (–2, 16) Point B: (–1, 4) Point C: (0, 1) For the points when x is positive, the y-values are fractions. Even though they are not integers, they still lie exactly on the curve. However, if you insist on using only points with integer coordinates, you might prefer to use the transformed form by considering –x as the new variable. In many classroom contexts, plotting (–2, 16), (–1, 4), and (0, 1) is enough to get the basic shape. Step 4. Describe the graph. • For x > 0: As x increases, f(x) = (1/4)^x gets smaller. In fact, f(1) = 1/4, f(2) = 1/16, and so on. The graph approaches the x–axis (y = 0) but never touches it. • For x < 0: As x decreases, f(x) becomes large. For example, f(–3) = 4^3 = 64, so the graph rises steeply to the left. • The curve always remains positive (f(x) > 0 for all x). Step 5. Graph by moving the key points. 1. Plot the point C at (0,1), which is the starting position. 2. Move one unit left from (0,1) to (–1, ?). Since f(–1) = 4^1 = 4, place the point at (–1,4). 3. Move one more unit to the left from (–1,4) to (–2, ?). Compute f(–2) = 4^2 = 16, so place the point at (–2,16). 4. To complete the graph, you could also plot additional points on the right side: for example, when x = 1, f(1) = (1/4)^1 = 0.25 (even though 0.25 is not an integer, you can mark it accurately on the graph); and when x = 2, f(2) = 1/16. 5. Draw a smooth curve passing through these points. On the right side (x > 0) the curve falls gradually toward the x–axis. On the left side (x < 0) the curve rises steeply. Step 6. Final look. Your graph should have a horizontal asymptote at y = 0 and should be decreasing as x increases. Summary of key points you can move: (–2, 16), (–1, 4), (0, 1), (1, 0.25), and (2, 0.0625). Even if you only use the three points with integer y–values ((–2,16), (–1,4), and (0,1)), you can then “fill in” the rest of the curve by understanding the transformation given by 4^(–x). Thus, by plotting these points and drawing a smooth curve through them, you have successfully graphed f(x) = (1/4)^x. To graph $$ f(x) = \left(\frac{1}{4}\right)^x $$, plot the key integer points: - (–2, 16) - (–1, 4) - (0, 1) Then, draw a smooth curve through these points, ensuring it approaches the x-axis as x increases and rises steeply as x decreases.

Question Graph the function \( f(x)=\left(\frac{1}{4}\right)^{x} \) by moving the key points. You can only graph integer points, for example you cannot plot the point \( \left(-1, \frac{1}{4}\right) \).

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