Question Video: Finding the Value of the Unknown Coefficients of a Linear Function given Its 𝑦-Intercept and Its Value at a Certain π‘₯ Value

Find π‘Ž and 𝑏, given 𝑓(π‘₯) = π‘Žπ‘₯ + 𝑏, where 𝑓(11) = 163 and the function intersects the 𝑦-axis at the point (0, βˆ’24).

02:29

Video Transcript

Find π‘Ž and 𝑏 given that 𝑓 of π‘₯ equals π‘Žπ‘₯ 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 π‘₯ equals π‘Žπ‘₯ 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 π‘₯ equals π‘Žπ‘₯ plus 𝑏, looks very similar to the form 𝑦 equals π‘šπ‘₯ plus 𝑏. In the form 𝑦 equals π‘šπ‘₯ plus 𝑏, the π‘š-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 π‘Ž-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 π‘₯ two minus π‘₯ 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 π‘₯ equals 11, 𝑓 of π‘₯ equals 163. From there, we can label these points π‘₯ one, 𝑦 one; π‘₯ two, 𝑦 two. And that means our π‘Ž-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 π‘Ž-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 π‘₯ is equal to 17π‘₯ minus 24, where π‘Ž equals 17 and 𝑏 equals negative 24.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.