Quadratic Graphs
Recognise that quadratic functions produce curved (parabolic) graphs, distinguish them from linear graphs, and use plotted quadratic graphs to estimate values and find approximate solutions
What a learner can do afterwards
- Explain why x² produces a U-shaped curve rather than a straight line
- Plot a simple quadratic such as y = x² − 4 from a table of values
- Read approximate solutions from a quadratic graph (e.g., where y = 0)
The lesson
You already know that graphs like y = 2x + 1 make a straight line. A quadratic graph is different because the x gets squared, like in y = x². Squaring changes the shape of the graph completely.
Look at y = x² for x = -2, -1, 0, 1, 2. The y-values are 4, 1, 0, 1, 4. They drop down to 0, then climb back up on the other side in the same pattern. That drop-and-rise pattern is what makes the graph curve into a U instead of running straight.
Here's the tell: a straight-line graph keeps the same steepness all the way along. A quadratic graph gets steeper the further you move from the middle, and it curves back upward on both sides. If a graph curves like a bowl, it's quadratic, not linear.
You can plot a quadratic from a table of values, just like you did with straight lines. For y = x² - 4, work out y for each x: x = -3 gives y = 5, x = -2 gives y = 0, x = -1 gives y = -3, x = 0 gives y = -4, x = 1 gives y = -3, x = 2 gives y = 0, and x = 3 gives y = 5.
Plot those points and join them with a smooth curve, and you get a U-shape that dips to its lowest point at x = 0. The curve crosses the x-axis, where y = 0, at x = -2 and at x = 2. Reading those crossing points off the graph gives you the approximate solutions, without any extra working out.
A quadratic graph curves into a U because squaring makes y rise again as x moves away from zero either way, and reading where it crosses zero gives you the solutions.
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8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.