➕ Algebra 1 · Quadratics
Graphing Quadratic Functions: fun ways and memory help
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
The basketball arc
A player lofts a basketball toward the hoop. It climbs, slows its climb, reaches its highest point, and then drops. Look at the height only. In a simplified model with no air and no spin, the ball’s height a short time before the top equals its height the same short time after the top. For example, 0.3 seconds before the peak and 0.3 seconds after it, the ball is at the same height.
How it connects: The height model is a quadratic in the time t with a < 0, so its graph opens down. The top of the arc is the vertex, and the maximum height happens at t = −b/(2a). The matching heights before and after the top are mirror points on either side of the axis of symmetry. Limit: this is a model. Air and spin change a real shot. Also, the parabola is the graph of height against time, so its horizontal axis shows time, not distance across the court.
The platformer jump
In a made-up platform game, the hero’s height above the ground, in blocks, is h = −4t2 + 8t at t seconds after the jump starts. At t = 0 the height is 0. At t = 1 it is 4 blocks, and at t = 2 the hero is back on the ground. At t = 1/2 and at t = 3/2, the height is 3 blocks both times.
How it connects: Since a = −4 is negative, the jump arc opens down. The axis of symmetry is t = −8/(2 · (−4)) = 1, so the vertex (1, 4) is the top of the jump. Take-off (t = 0) and landing (t = 2) are the zeros, and they are mirror images across the axis. The times t = 1/2 and t = 3/2 are each 1/2 away from the axis, so the heights match. Limit: the game is invented, and the formula describes only the time in the air. The parabola keeps going on both sides, but the hero is not there.
Meet halfway
Two friends live along one straight road, one at mile marker 4 and one at mile marker 12. They agree to meet at the point exactly halfway between their homes, at mile marker 8. Each of them walks 4 miles.
How it connects: When a parabola crosses the x-axis at two points, its axis of symmetry runs exactly halfway between them. For y = (x − 4)(x − 12), the x-intercepts are 4 and 12, the axis is x = (4 + 12)/2 = 8, and the vertex is (8, −16). Limit: this shortcut needs two x-intercepts. A parabola with none still has an axis, and then you use −b/(2a). Also, halfway tells you only the vertex’s x-coordinate. You still substitute to find its y-coordinate.
The lemonade stand
In a made-up rule, a stand that charges p dollars per cup sells 30 − 5p cups. Revenue is the money collected before any costs: R = p(30 − 5p) = −5p2 + 30p dollars. At 1 dollar per cup the stand sells 25 cups and collects 25 dollars. At 2 dollars it sells 20 cups and collects 40. At 3 dollars: 15 cups, 45 dollars. At 4 dollars: 10 cups, 40 dollars. At 5 dollars: 5 cups, 25 dollars. At 6 dollars: 0 cups, 0 dollars.
How it connects: The table is symmetric: 2 and 4 dollars give the same revenue, and so do 1 and 5. The axis is p = −30/(2 · (−5)) = 3, and the vertex (3, 45) says that the best price in this model is 3 dollars, with a maximum revenue of 45 dollars. The zeros p = 0 and p = 6 are the prices that bring in no money. Limit: the rule is invented, and real customers are not that tidy. Revenue is not profit, because costs are left out. And only prices from 0 to 6 dollars make sense here, so the domain in the story is smaller than all real numbers.
What goes up must come down
The proverb says that whatever rises will eventually fall back.
How it connects: A parabola that opens down matches this: it rises to the vertex and then falls, and the vertex is the top. The proverb is only a general saying, though. It does not say how high or when, and it does not say that the fall mirrors the climb. The parabola does: in the simplified ball model, the ball is at the same height at times equally far before and after the top, and the vertex tells you exactly when the top happens and how high it is. Limit: the saying does not fit every graph. After its vertex, a parabola that opens up keeps rising forever.
Looking up
Why was the upward-opening parabola so optimistic at its vertex? From there on, things were only looking up.
How it connects: Reading left to right, a parabola that opens up goes down until the vertex and then goes up, so its vertex is the lowest point and the y-values only increase after it. Limit: the pun works only for a parabola that opens up. A parabola that opens down reaches its vertex at the top, and after it the y-values only decrease.
Memory help
AVOID
AVOID guessing when you graph y = ax2 + bx + c. Axis: x = −b/(2a). Vertex: put that x back in to get y. Opens: up if a > 0, down if a < 0. Intercepts: plot (0, c) and its mirror point, and find the x-intercepts if there are any. Draw one smooth curve through the points.
Over two a
“Over two a goes the opposite of b: that’s the x where the vertex will be.” The opposite of b means change its sign: if b = −6 use 6, and if b = 6 use −6. Then divide by the whole of 2a.
Smile or frown
Picture a face. When a is positive the parabola smiles (opens up), and when a is negative it frowns (opens down). Only the sign of a draws the face. A positive b or c does not make it smile.
Fold the paper
Sketch a parabola on paper and fold the paper so the two halves land on top of each other. The crease is the axis of symmetry, and the crease touches the curve at the vertex. Every point on one half lands on a matching point on the other half, at the same height.
The sign flip on the axis
For y = x2 + 6x + 2, b = 6 and the axis is x = −6/2 = −3, not 3. When a and b have the same sign, the axis is to the left of the y-axis (negative x). When they have opposite signs, the axis is to the right of it. Check the sign before you plot.
A vertex needs two numbers
The vertex is a point, written (h, k). The maximum or minimum is only the y-value, k. If the vertex is (5, 7) and the parabola opens down, the maximum value is 7, and it occurs at x = 5.