 In many cases, we want to construct a function with particular properties. For example, a line with a specific slope, a parabola with a specific vertex, two graphs that intersect in specific places. To do this, it's helpful to remember every point x, y on the graph of an equation will make the equation true. For example, let's find b so that the point 3, 1 is on the graph of y equals x cubed plus bx squared plus 11x minus 15. So we know that 3, 1 is on the graph and every point x, y on the graph of an equation will make the equation true. So we know that x equals 3, y equals 1 makes the equation true. So remember, equals means replaceable. So every place we see an x, we can replace it with 3 and every place we see a y, we can replace it with 1. So in our equation, we can replace and then solve for b. What about a different situation where we know where two graphs intersect? Let's see if we can find additional intersection points. So remember, if x, y is at the intersection of two curves, it makes both equations true. And since we know x and we don't have y, so what can we do now? Well, remember, every point x, y on the graph of an equation will make the equation true. So x equals 1 does correspond to a point on the graph of y equals 3x minus 5. And since y equals 3x minus 5 and x is equal to 1, we can find y. And so the intersection point is that x equals 1, y equals negative 2, or at 1, negative 2. Now since 1, negative 2 is an intersection point, it makes both equations true. So we have y equals x cubed plus x squared plus cx plus 10 equals means replaceable. So we can replace x with 1 and y with negative 2. And that gives us an equation where the only thing we don't know is c. So solving for c and so we have our equations. Now we'd like to find any other intersection points. So we know that y equals 3x minus 5 and y equals x cubed plus x squared minus 14x plus 10 equals means replaceable. So any time we see y, about here, we can replace it with x cubed plus x squared minus 14x plus 10. And since this is a polynomial equation, let's get everything onto one side. And we have this nice cubic polynomial that we can try to solve. Now if you are walking down the street and this cubic equation fell out of the sky and hit you on the head, you should probably go to the emergency room to check out that concussion. But more importantly, we have ways we can solve cubic equations like this. But let's take advantage of the fact that we already know that x equals 1 is a solution because we know the two graphs intersect at x equals 1. And because we know x equals 1 is a solution, we know that x minus 1 is a factor. So we can factor the left hand side as x minus 1 times something. We can find the something by dividing. And we have a polynomial that we could try to factor. And now we have a product of three things equal to zero. So product equals zero only when one of them is equal to zero. This first factor, x minus 1, could be zero. That gives us the x equals 1 solution we already know. The other two factors give us x equals negative 5 and x equals 3 as additional solutions. And so solving, we find two more intersection points. A point with an x coordinate of negative 5 and a point with an x coordinate of 3. Well, technically these are half points. We need to know what y is. If x is equal to negative 5, we can use either equation to find y. Let's use the harder equation because we'd rather do more work than less work. What was that? Oh, maybe we'll use the easier equation. y equals 3x minus 5. We know what x is. And we can find our value of y, negative 20. And so that gives us negative 5, negative 20 is another intersection point. Similarly, if x equals 3, we can compute the value of y. And so 3, 4 is another intersection point.