12.3 Quadratic graphs and their features — Learn

Prerequisites: plotting coordinates, substituting values into expressions, and recognising roots of a graph.

Objectives: sketch a quadratic graph from a table or its factors, and identify roots, intercepts, and the turning point. This covers A11 and A12.

Shape and features

A graph of $y=ax^2+bx+c$ is called a parabola. If $a>0$, it opens upwards; if $a<0$, it opens downwards. The roots are the $x$-values where the graph crosses the $x$-axis. The turning point is its lowest or highest point.

Example 1. Find useful points for $y=x^2-4$.

Substitute values of $x$ and calculate $y$.

Plot the points. The graph opens upwards, has roots $x=-2$ and $x=2$, and turning point $(0,-4)$.

A guided graph investigation

Predict: for $y=(x-p)^2-4$, predict how changing $p$ affects the roots and turning point. Test: change $p$ in the figure and compare the plotted curve. Explain: describe what moved and what stayed the same.

Progress check

For the chapter model $h=-t^2+4t+5$, the coefficient of $t^2$ is negative, so the path opens downwards. Its roots will show when the ball is at ground level.

Watch out. The $y$-intercept occurs when $x=0$, not when $y=0$. Setting $y=0$ finds the roots. Check the axes before labelling a point.

You should now be able to

• sketch a quadratic graph using a value table;
• identify the roots, $y$-intercept, and turning point;
• describe how the sign of the squared-term coefficient affects the shape.