Question

Sketch the graph of the exponential equation. $$y=4(1.5)^{x}$$

Short answer

The graph of the function \(y=4(1.5)^{x}\) starts at the point (0,4) and curves upward as \(x\) increases, representing exponential growth.

Step-by-step solution

Identifying the parts of the function

The given function is \(y=4(1.5)^{x}\). Here, the base is 1.5, the coefficient of \(x\) is 1 (since it's not explicitly stated), and no constant is added or subtracted, thereby making 4 the initial value of \(y\). The function, therefore, represents an exponential growth.

Setting the x and y-axes

Before plotting the points and sketching the graph, set up an x and y-axes on the graph paper. Indicate the origin, which is the point (0, 0). This becomes the reference point from where the function starts.

Plotting the points

Start with x = 0 (for an exponential function, as x approaches -∞, y approaches 0, but never reaches it). So, when \(x=0\), using the function \(y=4(1.5)^{x}\), we get \(y = 4 * 1 = 4\). So, we plot the point (0, 4). Let's choose another point, for example, when \(x = 2\), then \(y = 4(1.5)^2 = 9\). So we plot the point (2, 9).

Sketching the graph

Join those points to get a curve and the graph. Extend the line to the x-axis but not crossing it (as the function will never become 0, it will just get closer). Likewise, as x grows positively, y tends to grow toward positive ∞.