Part 26 of 30 · QuadraticsSee the whole course
What is a quadratic function?
A quadratic function is a rule of the form y = ax² + bx + c, with a not equal to 0. Its graph is a smooth curve called a parabola: a bowl that opens up if a is positive, an arch that opens down if a is negative.
Where you see parabolas
A thrown ball, a jumping dolphin, the cable of a bridge, a satellite dish: they all follow parabolas. The x² in the rule is what bends the line. A line grows by the same amount every step. A parabola grows faster and faster: x² goes 1, 4, 9, 16, while 2x only goes 2, 4, 6, 8.
What a, b and c do
- a sets the shape. Positive opens up (a bowl), negative opens down (an arch). A big |a| makes it narrow, a small |a| makes it wide.
- c is where the parabola crosses the y-axis: the point (0, c). Put x = 0 and only c is left.
- b slides the turning point sideways, together with a. The turning point, the vertex, has x = −b ÷ 2a.
Example 1: y = x² − 4x + 3
a = 1, positive: a bowl.
c = 3: it crosses the y-axis at (0, 3).
Vertex: x = −(−4) ÷ (2 × 1) = 2, and y = 4 − 8 + 3 = −1, so the vertex is (2, −1).
It lands on the x-axis where x² − 4x + 3 = 0. That factors as (x − 1)(x − 3), so at x = 1 and x = 3.
The figure below starts with this one. Move a, b and c and watch the gold vertex, the orange y-intercept and the teal landing spots.
- a = 1: opens up, like a bowl
- c = 3: crosses the y-axis at (0, 3)
- vertex (2, −1), axis of symmetry x = 2
Example 2: y = −x² + 4
a = −1, negative: an arch. The top, the vertex, is at (0, 4).
It lands where −x² + 4 = 0, so x² = 4 and x = 2 or x = −2.
How to spot a quadratic in a table
For a line, the outputs go up by the same amount each step (see is it a linear function?). For a quadratic they do not, but the changes of the changes are constant. Take y = x² + 1:
x: 0, 1, 2, 3, 4
y: 1, 2, 5, 10, 17
differences: 1, 3, 5, 7
second differences: 2, 2, 2
The second differences are all the same, so it is a quadratic.
Watch it explained in under a minute
What is not a quadratic
- y = 2x + 3 has no x² term (a = 0), so it is a line.
- y = x³ + x² has an x³ term, so it is a cubic, not a quadratic.
The four mistakes to avoid
- Thinking a minus in front of x² moves the graph down. It flips it into an arch.
- Treating c as the height of the vertex. c is where it crosses the y-axis. They match only when b = 0.
- Drawing a V. A parabola is smooth and rounded at the vertex.
- Forgetting that a must not be 0. Without the x² term it is just a line.
Try it yourself
Tap a question to see the answer.
1. Which of these are quadratics: y = 3x² − 1, y = 2x + 5, y = x³ + x²?
Only y = 3x² − 1. The second is a line and the third is a cubic.
2. Describe y = −2x² + 8: which way it opens, the y-intercept, the vertex and the x-intercepts.
a = −2: it opens down. It crosses the y-axis at (0, 8), which is also the vertex because b = 0. For −2x² + 8 = 0, x² = 4, so the x-intercepts are x = 2 and x = −2.
3. Find the vertex and y-intercept of y = x² + 2x − 3.
Vertex x = −2 ÷ 2 = −1, and y = 1 − 2 − 3 = −4, so the vertex is (−1, −4). The y-intercept is (0, −3). It also factors as (x + 3)(x − 1), so it lands at x = −3 and x = 1.
Where this fits in
You have seen that a parabola is x² moved and reshaped, exactly the transformations from the last few posts. That gives a much easier way to find the vertex: vertex form.