Ex:  Find the Equation of a Quadratic Function from a Graph

We want to determine the equation of the quadratic function from the graph. We also want to give the function in general form. Remember, general form of a quadratic function is f of x equals ax squared plus bx plus c. But to find this form of the function, we're actually going to use the standard form first, which is f of x equals a times the quantity x minus h squared plus k. So the first thing we'll do is replace f of x with y. So we'll have y equals a times the quantity x minus h squared plus k. So in this form, hk would be the coordinates of the vertex, and xy would represent some point on the graph. So now let's take a look at our graph. This would be our vertex. Notice how the vertex would have an x-coordinate of negative 3 and a y-coordinate of positive 4. And then notice this point here on the function would have an x-coordinate of negative 4 and a y-coordinate of positive 2. This is actually all the information we need to get started. Since this point is the vertex, h equals negative 3 and k equals positive 4. And since this point is on the function, we can use x equals negative 4 and y equals positive 2. So again from here, we're going to substitute negative 3 for h, positive 4 for k, negative 4 for x, and positive 2 for y. Notice when we do this, we'll be left with one unknown, a. Once we determine the value of a, we'll have the function in standard form, which we'll then write in general form. So we're going to have 2 equals a times the quantity x minus h. Well, x is negative 4. Minus h is going to be minus negative 3. And we're going to square this and then add k, and k is positive 4. So now let's go ahead and simplify this. We'll have 2 equals a times, this is going to be negative 4 plus 3, or negative 1 squared plus 4. So we'll have 2 equals, negative 1 squared is just 1, so 1 times a is a, plus 4. Subtract 4 on both sides of the equation. This is 0, so we have negative 2 equals a. So our function in standard form is going to be f of x equals a, which is negative 2. And then we'll have the quantity x minus h squared. Remember, we do want x in our function, so we're going to have x minus h, but h is negative 3. Minus negative 3 will be plus 3. So we have x minus 3 squared, and plus k. Again, k is 4, so we have plus 4. But since we want this in general form, we'll have to multiply this out, combine like terms, and then put the terms in descending order, so it's in general form. So let's do that on the next slide. So we'll start by squaring the quantity x plus 3. Remember, there are no shortcuts. We'll have to write that out as 2 factors, and then multiply the 2 binomials. So we'll have 4 products, 1, 2, 3, and 4. We'll have negative 2 times this product, which will be x squared. Then we'll have plus 3x plus 3x, that's plus 6x, plus 9, plus 4. Now we'll distribute the negative 2. So we'll have negative 2x squared minus 12x minus 18 plus 4. And we have two like terms here that we can combine. So the function in general form would be f of x equals negative 2x squared minus 12x, and then negative 18 plus 4 is going to be negative 14, so we have minus 14. So this would be the general form of our quadratic function given by this graph. I hope you found this helpful.