The function f(x) = 4(2)x is an exponential growth curve that crosses the y-axis at (0, 4) and rises steeply to the right. If you’re choosing from multiple graphs, look for one with a y-intercept of 4, a curve that increases as it moves right, and a line that flattens toward (but never touches) the x-axis as it moves left.
Key Features of This Graph
The function f(x) = 4(2)x follows the general form f(x) = a · bx, where a is the initial value and b is the base. Here, the initial value is 4 and the base is 2. Because the base is greater than 1, this function models exponential growth, meaning the curve always increases from left to right.
The y-intercept is the easiest feature to check. Plug in x = 0: f(0) = 4(2)0 = 4(1) = 4. So the graph must pass through the point (0, 4). Any graph that crosses the y-axis at a different point is wrong, no matter how similar its shape looks.
From there, you can verify a couple more points. At x = 1, f(1) = 4(2)1 = 8. At x = 2, f(2) = 4(2)2 = 16. At x = −1, f(−1) = 4(2)−1 = 2. The correct graph passes through (−1, 2), (0, 4), (1, 8), and (2, 16).
The Shape You’re Looking For
Exponential growth curves have a distinctive “J” shape. On the left side of the graph (negative x-values), the curve gets closer and closer to the x-axis but never actually reaches it. The x-axis acts as a horizontal asymptote, meaning the function’s output approaches zero but stays positive. On the right side (positive x-values), the curve climbs sharply upward, getting steeper with each step.
This shape rules out several common distractors. A straight line is linear, not exponential. A U-shaped curve is a parabola (quadratic). A curve that drops from left to right is exponential decay, which happens when the base is between 0 and 1. The correct graph for f(x) = 4(2)x only goes up as you move right.
How to Eliminate Wrong Answers
Most multiple-choice versions of this problem include graphs that are designed to look similar. Here’s how to tell them apart quickly:
- Check the y-intercept first. If a graph crosses the y-axis at 1 or 2 instead of 4, it belongs to a different function. A basic f(x) = 2x without the coefficient crosses at (0, 1). A graph of f(x) = 2(2)x crosses at (0, 2). Only f(x) = 4(2)x crosses at (0, 4).
- Check the direction. The curve must rise from left to right. A graph that falls from left to right represents decay, such as f(x) = 4(0.5)x.
- Check steepness with a second point. At x = 1, the value should be 8. If the graph shows a value of 4 at x = 1 or some other number, it’s the wrong function.
Domain and Range
The domain of f(x) = 4(2)x is all real numbers. You can plug in any value of x, positive, negative, or zero, and get a valid output. On a graph, this means the curve extends infinitely in both horizontal directions.
The range is all positive real numbers, or (0, ∞). The function never outputs zero or a negative number. This is visible on the graph: the curve hugs the x-axis on the left side but never dips to or below it.
Why the 4 Matters
The coefficient 4 in front of 2x acts as a vertical stretch. Compare f(x) = 2x and f(x) = 4(2)x: they have the same base and the same general shape, but every output of the second function is four times larger. The basic curve f(x) = 2x passes through (0, 1), (1, 2), and (2, 4). Multiplying by 4 shifts those points to (0, 4), (1, 8), and (2, 16). The curve looks the same but sits higher on the coordinate plane and appears to grow faster, even though the rate of doubling is identical.
This distinction is the most common source of confusion when picking between similar-looking graphs. Two exponential curves can have the same shape and direction but different y-intercepts. The y-intercept always equals the coefficient a in f(x) = a · bx, so it’s the fastest way to match a function to its graph.