Question Video: Forming the Equation of an Exponential Function from Its Graph Mathematics

Observe the given graph, and then answer the following questions. Find the 𝑦-intercept in the shown graph. As this graph represents an exponential function, every 𝑦-value is multiplied by 𝑏 when π‘₯ increases by Ξ”π‘₯. Find 𝑏 for Ξ”π‘₯ = 1. Find the equation that describes the graph in the form 𝑦 = π‘Žπ‘^(π‘₯/Ξ”π‘₯).

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Video Transcript

Observe the given graph and then answer the following questions. Find the 𝑦-intercept in the shown graph. As this graph represents an exponential function, every 𝑦-value is multiplied by 𝑏 when π‘₯ increases by Ξ”π‘₯. Find 𝑏 for Ξ”π‘₯ equals one. Find the equation that describes the graph in the form 𝑦 equals π‘Žπ‘ to the power of π‘₯ over Ξ”π‘₯.

Let’s begin by finding the graph’s 𝑦-intercept. We can see that the line of the graph passes through the point zero, 10. Therefore, the 𝑦-intercept is 10. Now, let’s find the value of 𝑏 in the equation of the graph. In the problem, we are told that the graph represents an exponential function for which every 𝑦-value is multiplied by 𝑏 when π‘₯ increases by Ξ”π‘₯. And we’re asked to find 𝑏 when Ξ”π‘₯ equals one. We’ve already seen that the graph passes through the point zero, 10. And we can also see that it passes through the point one, 20. The change in π‘₯ between these two points is one, so it satisfies Ξ”π‘₯ equals one.

We are told that the 𝑦-value is multiplied by 𝑏 when π‘₯ increases by Ξ”π‘₯. If the 𝑦-value of our first point is 𝑦 one and the 𝑦-value of our second point is 𝑦 two, then 𝑏 equals 𝑦 two over 𝑦 one which is equal to 20 divided by 10, which is equal to two. And finally, we need to find the equation that describes the graph in the form 𝑦 equals π‘Žπ‘ to the power of π‘₯ over Ξ”π‘₯. We are already given Ξ”π‘₯ equals one, and we have found the value of 𝑏 equal to two. Therefore, 𝑦 equals π‘Ž times two to the π‘₯ over one, which is just equal to π‘Ž times two to the π‘₯. To find the value of π‘Ž, we can substitute the π‘₯- and 𝑦-values of points we know to be on the graph.

We’ve already found the 𝑦-intercept to be 10, so we know that the point zero, 10 lies on the graph. This means we can substitute the values of π‘₯ equals zero and 𝑦 equals 10 into the equation which gives us 10 equals π‘Ž times two to the zero. By the zero-exponent rule, two to the zero is just equal to one. Therefore, this gives π‘Ž equals 10. This gives us our equation and the final answer 𝑦 equals 10 times two to the π‘₯.

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