Year 10: Parabolas and graph intersections

In Australian Year 10 Mathematics, the parabola is often your first curved graph in coordinate geometry. You need to recognise its shape, sketch it from a simple equation, and understand how to find where it intersects with another graph.

A parabola often comes from a quadratic rule such as

y = x^{2}

. Intersections are points where two graphs have the same x-value and the same y-value.

Recognising the shape

The graph of $y = x^{2}$ is U-shaped and symmetric. If the coefficient of $x^{2}$ is positive, the graph opens upward. If it is negative, it opens downward.

A downward-opening parabola with its vertex above the origin and a dashed axis of symmetry.

The sign in front of $x^{2}$ controls whether the parabola opens up or down, and the graph stays symmetric about its vertical centre line.

Example 1: Describe

y = - x^{2} + 4

The negative sign means the parabola opens downward.
The $+ 4$ moves the graph upward.
The graph has its highest point above the origin.

Sketching with a table of values

A safe Year 10 method is to choose a few x-values, calculate the corresponding y-values, and plot them. Using values on both sides of the centre helps show the symmetry clearly.

Points from a table of values plotted on the graph of y equals x squared minus one.

Plotting matching points on either side of the centre makes the symmetry visible and helps you draw the parabola as a smooth curve instead of disconnected segments.

Example 2: Sketch

y = x^{2} - 1

Substitute $x = - 2, - 1, 0, 1, 2$ .
The y-values are $3, 0, - 1, 0, 3$ .
Plot $(- 2, 3), (- 1, 0), (0, - 1), (1, 0), (2, 3)$ .
Join them smoothly to form the parabola.

Intersection of two graphs

If two graphs intersect, their y-values are equal at the same x-value. Algebraically, this means you set the equations equal and solve.

A line and a parabola crossing at two intersection points marked on the graph.

Intersections are the shared points where both rules produce the same coordinate, so the graph matches the algebraic equation you solve.

Example 3: Find the intersections of

y = x + 1

and

y = x^{2} - 1

Set the equations equal: $x + 1 = x^{2} - 1$ .
Rearrange: $x^{2} - x - 2 = 0$ .
Factor: $(x - 2) (x + 1) = 0$ .
So $x = 2$ or $x = - 1$ .
Substitute into $y = x + 1$ .
If $x = 2$ , then $y = 3$ .
If $x = - 1$ , then $y = 0$ .

Answer: the intersection points are

(2, 3)

and

(- 1, 0)

Link to inequalities

Once you can sketch lines and parabolas, you can reason about which region of the graph satisfies an inequality. That is why sketching and intersections support later work on systems of linear inequalities in Year 10.

A shaded region between a parabola and a line to illustrate graph inequalities.

Inequality questions are about choosing the correct region of the plane, not just locating the boundary curves.

Common mistakes

Plotting too few points to show the curved shape properly.
Forgetting that a parabola is symmetric.
Finding x-values for intersections but not calculating the matching y-values.
Drawing straight line segments instead of a smooth curve.

Study routine

For a parabola, start with a small table of values. For intersections, set the equations equal, solve carefully, and then substitute back. A quick sketch after the algebra is a strong final check.