Parent guide · 6 min read
Quadratic functions for kids: why the graph is a curve and what its key points mean
A linear function makes a straight line. A quadratic function, one with an x² term such as y = x² − 4x + 3, makes a curve called a parabola. Children usually meet it in Algebra 1 (around age 14–15) and find it harder than lines because there is more to read off the graph: two crossing points, a turning point and a symmetry line. This guide shows what each piece means, a table method that works every time, and three worked examples.
Try this tonight
4 things you can do at home
Make a table of x and y values
For y = x² − 4x + 3, choose x = 0, 1, 2, 3, 4. Work each out on its own line: y = 3, 0, −1, 0, 3. Doing the arithmetic one value at a time avoids the most common error, which is squaring and subtracting in the wrong order.
Plot the points and join them with a smooth curve
Use graph paper and draw a curve, not straight segments. Ask your child what they notice: the y-values go down and then back up, and the table reads the same on both sides of x = 2.
Name the three key features
On the graph mark where it crosses the x-axis (x = 1 and x = 3, the roots), the lowest point (2, −1), the vertex, and where it crosses the y-axis (y = 3). Labelling these in writing makes the vocabulary stick.
Check by substituting
Put x = 1 and x = 3 back into the equation and confirm y is 0. Checking one point takes ten seconds and tells your child whether the plot, not just the algebra, is right.
Why a quadratic curves when a linear function does not
In a linear function, each step of 1 in x changes y by the same amount, so the graph is straight. In y = x² − 4x + 3 the y-values are 3, 0, −1, 0, 3, so the changes are −3, −1, +1, +3. The change itself changes by 2 each time, and that steady "change of the change" is the signature of a quadratic.
That is why the graph bends. The y-values fall quickly, slow down at the bottom, then rise again at the same pace they fell. The two sides mirror each other across a vertical line called the axis of symmetry, here x = 2.
A worked example: reading roots, vertex and y-intercept
Take y = x² − 4x + 3. Setting y = 0 and factoring gives (x − 1)(x − 3) = 0, so x = 1 or x = 3. These are the roots, the two places the curve meets the x-axis. The vertex sits halfway between them, at x = 2, and y = 4 − 8 + 3 = −1, so the vertex is (2, −1). The y-intercept is where x = 0, which gives y = 3.
The same feature can be found from the table without any factoring: the zeros in the y column are the roots, and the smallest y is the vertex. Children who see that both methods agree trust the algebra more.
A worked example: a ball thrown upward
The height of a ball in metres after t seconds is h = −5t² + 20t. The negative −5 means the curve opens downward, so the vertex is a maximum. At t = 2, h = −20 + 40 = 20, so the ball peaks at 20 m after 2 seconds. It lands when h = 0: −5t(t − 4) = 0, so t = 0 (the throw) or t = 4 (the landing).
Word problems like this are where students most often lose marks, usually by giving the time of the peak when asked for the height. Writing "time" and "height" next to each answer fixes it.
What the number in front of x² does
Compare y = x², y = 2x² and y = −x². At x = 3 they give 9, 18 and −9. A bigger positive a makes the curve narrower and steeper, and a negative a flips it upside down. Changing c only slides the whole curve up or down: y = x² + 3 is y = x² lifted 3 units.
Spending ten minutes sketching these three on one set of axes teaches more than a page of rules, because the child sees what each letter controls.
Signs it’s time for outside help
When home help has done what it can
- Joins the plotted points of a quadratic with straight lines instead of a smooth curve.
- Cannot say what the roots, vertex and y-intercept are on a graph.
- Squares −3 and gets −9, or computes −x² and x² as if they were the same.
- Finds the roots but gives them as points like (1, 0) one time and as plain numbers the next, without knowing the difference.
- Can factor an expression on a worksheet but cannot connect it to where the graph crosses the axis.
- Mixes up the highest point and the time it happens in word problems.
Questions parents ask
FAQ
What grade do children learn quadratic functions?
In the US they are usually part of Algebra 1 or Algebra 2 (ages 14–16), covered by the Common Core high school functions and equations standards. In Australia, quadratic graphs are in the later secondary years.
What is the difference between a quadratic equation and a quadratic function?
A quadratic function, y = ax² + bx + c, describes a whole curve. A quadratic equation sets that expression to a specific value, usually 0, and asks which x-values work. The solutions of the equation are the roots of the function.
What is a parabola?
A parabola is the U-shaped curve that is the graph of a quadratic function. It is symmetrical, with a single turning point called the vertex.
Do all quadratics cross the x-axis?
No. A parabola can cross it twice, touch it once or miss it entirely. For example, y = x² + 1 has its lowest point at (0, 1), so it never reaches y = 0.
Should my child learn factoring before graphing quadratics?
Either order works, and many schools do both together. A table of values needs only arithmetic, so a child who is shaky on factoring can still graph and read the key points while they practise it.
Further reading
Classes that fit
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