Quadratic Graphs

A straight line first

The equation y = x says that y is always equal to x. Its points include (−3, −3), (−1, −1), (0, 0), (2, 2) and (3, 3). Each step of 1 to the right takes the line 1 up, the same amount every time, and that is why its graph is a straight line through the origin.

The points from (−3, −3) to (3, 3) lie on the straight line y = x, which rises 1 for every 1 across, all the way along.

Square every x

Now square each input instead: $y=x^2$y = x². For x = 0, 1, 2 and 3 the values of y are 0, 1, 4 and 9. For the negative inputs, a negative number times itself is positive, so ${\left(-1\right)}^2=1$(−1)² = 1, ${\left(-2\right)}^2=\left(-2\right)\times \left(-2\right)=4$(−2)² = (−2) × (−2) = 4 and ${\left(-3\right)}^2=9$(−3)² = 9. Negative inputs come back positive, and x and −x always give the same y.

The steps up are no longer equal. From x = 0 to 1, y rises by 1; from 1 to 2 it rises by 3; from 2 to 3 it rises by 5. Each step is steeper than the one before, so the points cannot lie on a straight line. Join them with one smooth curve.

The seven points of $y=x^2$y = x², from (−3, 9) to (3, 9), joined by one smooth curve. The straight line y = x is drawn beside it.

The vertex and the axis of symmetry

Read the curve from left to right. It falls, from 9 to 4 to 1, turns at (0, 0), then climbs through 1, 4 and 9. The point where it turns is the turning point, also called the vertex. For $y=x^2$y = x² it is the lowest point of the graph, because a square is never negative, so y is never less than 0.

The curve is symmetric about the y-axis, because x and −x give the same value of y: fold the graph along the y-axis and the two halves fit. The line of the fold is the axis of symmetry, and it passes through the vertex. A curve of this shape is called a parabola.

The curve turns at its vertex (0, 0). The points (−2, 4) and (2, 4) are the same height, one on each side of the axis of symmetry, x = 0.

The shape of $y=a{x}^2+bx+c$y = ax² + bx + c

Any equation $y=a{x}^2+bx+c$y = ax² + bx + c, where a is not 0, has a parabola for its graph. The number a decides which way it opens. When a is positive, as in $y=x^2$y = x², the parabola opens upward and the vertex is the lowest point. When a is negative, as in $y=-x^2$y = −x², every value of y is the negative of its value for $y=x^2$y = x², so the parabola is reflected in the x-axis: it opens downward, and the vertex is the highest point.

The number c is the value of y at x = 0, so the parabola crosses the y-axis at (0, c). In $y=x^2-4$y = x² − 4, every value of y is 4 less than in $y=x^2$y = x², so the whole curve moves 4 down: the vertex is (0, −4), and the curve crosses the x-axis where $x^2-4=0$x² − 4 = 0, at x = −2 and x = 2.

Symmetry finds the vertex

Take $y=x^2-6x+5$y = x² − 6x + 5. It crosses the x-axis where y = 0. Factorize: $x^2-6x+5=\left(x-1\right)\left(x-5\right)$x² − 6x + 5 = (x − 1)(x − 5), so y = 0 at x = 1 and at x = 5. These values of x are the roots.

The axis of symmetry is halfway between the two roots: $x=\left(1+5\right)/2=3$x = (1 + 5) / 2 = 3. The vertex lies on the axis, so put x = 3 into the equation: $y=3^2-6\times 3+5=9-18+5=-4$y = 3² − 6 × 3 + 5 = 9 − 18 + 5 = −4. The vertex is (3, −4), the lowest point, because a = 1 is positive.

Any two points at the same height are also symmetric about the axis. At x = 0 the curve is at y = 5, and at x = 6 it is at $y=6^2-6\times 6+5=36-36+5=5$y = 6² − 6 × 6 + 5 = 36 − 36 + 5 = 5 as well, so the axis is halfway between them too: (0 + 6) ÷ 2 = 3. When a curve does not cross the x-axis, a pair of equal heights finds the axis instead.

$y=x^2-6x+5$y = x² − 6x + 5 crosses the x-axis at 1 and 5 and is at height 5 at x = 0 and x = 6. Both pairs are centered on x = 3, where the vertex is (3, −4).

The two handles are the roots of $y=\left(x-r_1\right)\left(x-r_2\right)$y = (x − r₁)(x − r₂). Move either root and the parabola follows, with its axis of symmetry always at half their sum, halfway between them.

Reading a parabola in a real problem

When a parabola describes something real, such as the path of a jet of water, each feature of the graph answers a question. The roots are where the height is 0. The value of y at x = 0 is the height where the measuring starts, such as at a wall. The vertex gives the greatest or least value, and the x-coordinate of the vertex says where it happens.

Check each root against the situation. A root can be a correct solution of the equation and still describe a place the real object never reaches, such as a distance behind the wall a jet comes out of.