C3.1; E3.1 - Linear and quadratic graph shapes

Linear graphs

The rule $f(x)=2x+1$ gives outputs 1, 3 and 5 for inputs 0, 1 and 2. Equal increases in the input give equal increases in the output, so its graph is straight. The rule $g(x)=x^2+1$ gives 1, 2 and 5 at those inputs; its changing increase produces a curve.

A vertical line $x=k$ is a straight line but is not the graph of a function assigning a single $y$ to each $x$: one input would have many outputs. The phrase “linear function” here concerns $y=f(x)$.

[DIAGRAM: asset_name: Rising falling and constant linear functions; asset_slug: graph_shapes_linear; caption: The coefficient of x controls whether a straight graph rises, falls or stays horizontal.; description: Three separate accurately scaled coordinate grids show y equals two x plus one, y equals minus two x plus one and y equals one. Each equation is printed above its own graph and the common intercept zero comma one is marked. The two sloping lines rise and fall respectively, while the third line is horizontal.]

Change only the gradient's sign. The intercept stays in the same place.

The intercept alone cannot distinguish a line from a curve. Check the whole shape and the direction as well as where it crosses an axis.

Identify the shape using both coefficients.

Quadratic graphs

Squaring $-2$ and 2 gives the same result. This repeated pairing makes $y=x^2$ symmetric about the $y$-axis, with its lowest point at the origin. Adding a linear term can move the line of symmetry, but the quadratic curve still has one turning point.

The sign of $a$ decides whether the curve opens upwards or downwards. It does not decide whether the whole curve is above or below the $x$-axis. The constant $c=f(0)$ gives the $y$-intercept, which need not be the vertex.

[DIAGRAM: asset_name: Quadratic vertices and symmetry; asset_slug: graph_shapes_quadratic; caption: Both curves turn at x equals 1, but one vertex is a minimum and the other a maximum.; description: Two equal-scale coordinate grids show y equals x squared minus two x minus three and y equals minus x squared plus two x plus three. The first vertex is labelled one comma minus four and its intercepts are minus one comma zero, three comma zero and zero comma minus three. The second vertex is one comma four with intercepts minus one comma zero, three comma zero and zero comma three. Dashed vertical lines through x equals one mark the lines of symmetry.]