Find 𝑎 and 𝑏 given that 𝑓 of 𝑥 equals 𝑎𝑥 plus 𝑏, where 𝑓 of 11 is equal to 163 and the function intersects the 𝑦-axis at the point zero, negative 24.
Let’s think about what we know. We’re working with a function in the form 𝑓 of 𝑥 equals 𝑎𝑥 plus 𝑏. We know that 𝑓 of 11 is equal to 163. And we also know that this function intersects the 𝑦-axis at the point zero, negative 24. We should recognize that the function we’re working with, 𝑓 of 𝑥 equals 𝑎𝑥 plus 𝑏, looks very similar to the form 𝑦 equals 𝑚𝑥 plus 𝑏. In the form 𝑦 equals 𝑚𝑥 plus 𝑏, the 𝑚-value is the slope of the function and the 𝑏-value will be equal to the 𝑦-intercept, where the function intersects the 𝑦-axis. And it’s always located at the point zero, 𝑏.
Looking at our equation, the 𝑎-value will be equal to the slope and the 𝑏-value will be equal to the 𝑦-intercept. If we know two points, we calculate the slope by saying 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. At first, it might not seem like we have two points. However, we can rewrite this 𝑓 of 11 equals 163 into coordinate form. When 𝑥 equals 11, 𝑓 of 𝑥 equals 163. From there, we can label these points 𝑥 one, 𝑦 one; 𝑥 two, 𝑦 two. And that means our 𝑎-value will be equal to negative 24 minus 163 over zero minus 11.
Negative 187 over negative 11 equals positive 17. And that means our 𝑎-value is equal to 17. The key to finding our 𝑏-value is to recognize that the 𝑏-value is the 𝑦-intercept. And we’ve been given the 𝑦-intercept. Since we know this function crosses the point zero, negative 24, the 𝑦-intercept, 𝑏, must be equal to negative 24. Our function would look like this. 𝑓 of 𝑥 is equal to 17𝑥 minus 24, where 𝑎 equals 17 and 𝑏 equals negative 24.