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Find an expression for the function whose graph is the given curve.

The line segment joining the points $ (1 , -3) $ and $ (5 , 7) $

$f(x)=\frac{5}{2} x-\frac{11}{2}, 1 \leq x \leq 5$

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All right, let's find the function that is the line between these two points. So in order to get the equation of a line, we need the slope and the Y intercept. So to get the slope, we can use our slope formula. Why? To minus y 17 minus negative three over X two minus x 15 minus one. And that gives us 14th which reduces to five house. And now we confined the Y intercept. So let's use point slope form, which is why minus y one equals M times X minus X one. And we could use either one of our two points. I'm just going to use the 2nd 1 so we have y minus seven equals five halves times a quantity X minus five. If we want to change this equation into slope intercept form, we can start by distributing the five halves. So why minus seven equals five halves X minus 25 halves, and then we can add seven to both sides, adding seven is like adding 14 halves, so we end up with y equals five halves minus 11 house. So that's the equation of the line through those two points. And because we just want this segment, we're going to restrict the domain and we're only going to go from X equals 12 x equals five.