Parabolas: Vertex, Axis of Symmetry, Intercepts

In this lesson you will: Locate the axis of symmetry from the coefficients, then substitute to get the vertex; Read the y-intercept off the constant term and the x-intercepts off the factors; Decide from the sign of a alone whether the vertex is the lowest point or the highest.

A parabola is a mirror image of itself

Every graph of y=ax^2+bx+c has one vertical line it folds onto perfectly. That line is the axis of symmetry, and finding it first makes everything else cheap: the vertex sits on it, the two x-intercepts sit at equal distances either side of it, and every point you plot hands you a second point for free.

The axis comes from the coefficients, not from the graph. You never have to draw anything to find it.

Four features, in the order that costs the least work

Take f(x)=x^2-6x+5. The axis is x=-(-6)/(2(1))=3. Substituting, f(3)=9-18+5=-4, so the vertex is (3,-4).

The y-intercept needs no work at all — it is the constant term, because f(0)=c. The x-intercepts come from the factoring you did in Chapter 15: x^2-6x+5=(x-1)(x-5), so the graph meets the horizontal axis at 1 and at 5.

Every point you plot is really two points

The y-intercept of f(x)=x^2-6x+5 is (0,5). The input 0 is three units left of the axis x=3, so three units right of the axis — at x=6 — the output must be 5 again. Check it: f(6)=36-36+5=5.

That is two plotted points for one substitution. On a sketch where you are allowed four or five points, symmetry halves the arithmetic.

Common mistake — Dropping the minus sign in the axis formula

For f(x)=x^2+8x+7 a student computes (8)/(2)=4 and reports the axis of symmetry as x=4.

The slip is easy to make because the formula is usually read aloud as \"b over two a\" and the minus sign in front lives outside the fraction, where the eye skips it.

The axis is x=-(8)/(2(1))=-4. The two answers are not close: f(4)=16+32+7=55, while f(-4)=16-32+7=-9. One is the bottom of the curve; the other is far up the right-hand branch.

How to catch it: The axis must land between the x-intercepts. Here x^2+8x+7=(x+7)(x+1), so the zeros are -7 and -1, and their average is -4. If the number you wrote is not between the zeros, you lost the sign.