Give the equation of each exponential function whose graph is shown.

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Identify the general form of an exponential function: \(y = a \cdot b^x\), where \(a\) is the initial value (the value when \(x=0\)) and \(b\) is the base or growth factor.

For the first graph, use the point where \(x=0\) to find \(a\). Since the point is \((0, 2)\), substitute to get \(a = 2\). So the function starts as \(y = 2 \cdot b^x\).

Use another point from the first graph, for example \((1, 4)\), and substitute into the equation: \(4 = 2 \cdot b^1\). Solve for \(b\) by dividing both sides by 2, giving \(b = 2\).

Write the equation for the first graph as \(y = 2 \cdot 2^x\) after finding \(a\) and \(b\).

Repeat the process for the second graph: start with the point \((0, 1)\) to find \(a = 1\), so \(y = 1 \cdot b^x = b^x\). Use the point \((1, 4)\) to find \(b\) by substituting: \(4 = b^1\), so \(b = 4\). The equation for the second graph is \(y = 4^x\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Exponential Function Form

An exponential function is generally written as y = ab^x, where 'a' is the initial value (y-intercept) and 'b' is the base or growth factor. Understanding this form helps in identifying the equation from given points on the graph.

Using Points to Find Parameters

Given points on the graph, especially the y-intercept (where x=0), you can find 'a' directly. Then, using another point, substitute x and y values to solve for the base 'b'. This process is essential to determine the exact equation of the exponential function.

Graph Interpretation and Growth Behavior

The shape of the graph shows exponential growth if it rises rapidly as x increases. Recognizing this behavior confirms the base 'b' is greater than 1. The plotted points (0,2), (1,4), and (2,8) indicate doubling behavior, which helps in identifying the base.