Intercept Form
The axis of symmetry lies directly between the two roots
The next form we'll go over is intercept form, y = a(x- p)(x - q). This is the general form, and here are a few specific examples: y = -(x - 1)(x + 5) and y = 3(x + 5)(x + 9).
While it is true that every once in awhile you'll be given a problem that's already in intercept form, it will often be the case that you'll have to first factor the standard-form equation to make it look like intercept form. Although this can sometimes be a headache, there are advantages to doing the work. The a value will, again, tell you whether the parabola is concave up or down, and if you want to find the y-intercept, you can simply substitute in x=0 and quickly evaluate a(-p)(-q).
Where intercept form gets its name and passes standard form in usefulness, is in its ability to not just tell you where the y-intercept is but also where the x-intercepts are. Because the x-intercepts are where y=0, substituting in either p or q will give you a zero in your product, turning the entire equation into zero. Therefore, p and q are the two x-intercepts, or roots, of your quadratic. Be careful with the signs on your roots, though. Because the general equation has a -p and -q, an (x - 5) would actually mean a root at x=5, while an (x + 5) would mean a root at x= -5.
Lastly, because parabolas are symmetrical, the axis of symmetry must lie directly in between the two roots. This means you can find it on your graph by working your way into the middle or algebraically by calculating the average between the two points: x = (p + q)/2.
Vertex Form
kkk
The h and k values represent the vertex of the parabola
And finally we come to vertex form: y = a(x - h)2 + k. This is the general form, and these are some specific examples: y = 9(x + 5)2 - 1 and y = -(x - 3)2 - 1.
This time, getting your quadratic into this form requires you to complete the square, which is possibly the hardest algebraic trick of them all. But if you can, you are going to be rewarded for your hard work. First off, the a value still tells us whether it's concave up or down, and the y-intercept is still easily found by substituting in x= 0 and evaluating. But now, just like intercept form gave us the intercepts, vertex form will give us the vertex of our parabola straight from the equation: h is going to become the x-coordinate, and k will become the y-coordinate, of our vertex. Now, we can easily tell where the axis of symmetry is simply by remembering that it goes right through the middle of the graph where the vertex is. Therefore, the axis of symmetry is just the line x = h.
The h and k values represent the vertex of the parabola
And finally we come to vertex form: y = a(x - h)2 + k. This is the general form, and these are some specific examples: y = 9(x + 5)2 - 1 and y = -(x - 3)2 - 1.
This time, getting your quadratic into this form requires you to complete the square, which is possibly the hardest algebraic trick of them all. But if you can, you are going to be rewarded for your hard work. First off, the a value still tells us whether it's concave up or down, and the y-intercept is still easily found by substituting in x= 0 and evaluating. But now, just like intercept form gave us the intercepts, vertex form will give us the vertex of our parabola straight from the equation: h is going to become the x-coordinate, and k will become the y-coordinate, of our vertex. Now, we can easily tell where the axis of symmetry is simply by remembering that it goes right through the middle of the graph where the vertex is.
Therefore, the axis of symmetry is just the line x = h.