Find an equation of the tangent line to the graph of $ y = g(x) $ at $ x = 5 $ if $ g(5) = -3 $ and $ g'(5) = 4 $.
All right, we have a function Y equals G of X. Let's just write it off over here. We have a function Y equals G of X. We do not have the graph of G of X. But we do know a little bit of information. We know that G 05 is negative three. uh that means when X is five, the value of the function G is negative three. That means 2.5 common negative three is on the graph. So that's this point right here. All right. Here's the .5 comma -3. We know this point is on the graph of G Fx. Now we also know the derivative of G Uh at x equals five is 4. So the derivative of a function At a particular x value. Like five means that the tangent line to the curve to the graph of the function at X equals five is going to have the same slope as the derivative. That's the most important point. So, let me rephrase that the slope of a line that is tangent to a graph At X equals five will be the derivative of the function at five. So, when G prime of five is equal to four, this means That four will be the slope of the tangent line to slope of the tangent line. It's really pretty simple and try not to make it too complicated. The slope of the tangent line is the derivative of the function at the point where the tangent line touches the graph. So four is going to be the slope of the tangent line. So what is the tangent line going to look like? We're not actually even going to graph the function G Fx. I'm just going to graft the tangent line Now we know that when X is five to function G 05 equals negative three. So that's how we know that the 30.5 common negative three is on the graph of G Fx. That's the only part of G F X. That we're going to graph. But the tangent line detain geant line is going to touch the graph of G F X at this point. So the tangent line will pass through the 0.5 common negative three. And we know the slope of the tangent line is the same as the derivative. So G prime at five is four. So four will be the slope of the change of life. Now you need to remember that you can think of slope as rise over run so we can write this whole number four as 4/1. So, from this point since the slope of the tangent line which of course equals two derivative is 4/1. Rise over run. We're going to rise four and run one. So rising four units from this 40.1234. Okay. And then running one brings us right here. So let's draw a line from this point. Rise four over one. That's what the tangent line looks like. So this black line is not the graph of g F X. It's the graph of the tangent line lastly what we need to do. We just need to write the equation of the tangent line. Now in algebra, you learned what was called the point slope form of the equation of a line was why minus Y. One equals M Times X -X one. The Y and E X are going to stay Y and X. But we're going to substitute in for X one, Y one and The X one is going to be ah the coordinates of this point. Okay, this point is tangent to the function G. Of X. Okay, this line is tangent to the function G fx at this point. So even though we don't have the great G of X, we know what the tangent line looks like. And we need to coordinates uh of this point right here to be our X. One and ry one, those two points are on the tangent line. We are trying to come up with the equation of the tangent line. So you need a point on the tangent line. This red dot is red point is on the tangent line. And so these coordinates are X one and Y one. Lastly, we just need em while the slope of the tangent line, once again is the derivative of the function at the point. So the derivative G prime when x was five G prime is equal to four. And so now we just substitute in Why minus why one? Why subtract why one? Why subtract negative three equals M. M. The slope is the derivative for and then times X to Y. N E. X. Stay the same. You just substituting it for Y. One M. And now X one. Uh So X subtract now we substituted for X one. X one was five. Moving this down just a little bit. All right now we just do a little algebra to clean up oriental. Why should attract negative three? Uh Is why plus three Equals distributing the multiplication four times x is for X. Ah Subtract four times 5 is 20. Now we can attract this three from both sides of the equation and we get y equals four X. Subtract 23. That is the equation of our tangent line. So even though we never had a picture of the graph of G F X. Uh We just knew that this black line here, this this line was tangent. It touched the graph of G F X at this point. And we wanted to find the equation of this tangent line. And to find the equation of any line. You just need a point on the line. So we had this point that was on the tangent line and he needed to know the slope of the line. And of course the slope of a change in line is equal to the derivatives