Middle school maths, year 3 · unit 4
Quadratic Functions
Throw a basketball and it does not fly straight: it curves up and comes back down. Draw that path as a graph and you get a curve, not a line. The graph of a linear function was straight; once x is squared, the graph bends. We start by plotting points one at a time.
Plot the points, get a curve
A board to adjustWhat does the graph of y = x² look like? Change x and plot the points one by one.
a sets width and direction
A board to adjustHow do y = 2x² and y = −x² differ from y = x²? Drag the handle up and down.
Moving the vertex
A board to adjustMove the whole parabola: how does the formula change? Drag the gold vertex.
Through three points
A board to adjustWhat do the places where a parabola meets the x-axis mean? Drag the vertex so the curve passes through all three pins.
Try it yourself
Use what you found on the boards. The numbers change every time and the easy questions come first. Get three right and the lamp lights.
What to take away from this unit
- A function where y is a quadratic in x is a quadratic function, and its graph is a parabola. The line it folds onto itself along is the axis, and the point where it meets the axis is the vertex.
- y = ax²: it opens upward if a is positive, downward if negative. The larger the size of a, the narrower the graph.
- y = a(x − p)² + q is y = ax² moved p along the x-axis and q along the y-axis. Vertex (p, q), axis x = p.
- Moved 3 to the right it is x − 3. The vertex sits at the x that makes the bracket 0.
- y = ax² + bx + c is the same kind of parabola. c is the y-coordinate where it meets the y-axis.
- The x-coordinates where the parabola meets the x-axis are the solutions of ax² + bx + c = 0. The axis runs midway between those two points.