Quadratic functions and their graphsEdexcel International A Level Maths: Revision notes

A quadratic function has the form $f(x)=ax^2+bx+c$ with $a\neq0$. Its graph is a smooth symmetrical curve called a parabola.

• If $a>0$ the graph is $\cup$-shaped with a minimum point.
• If $a<0$ the graph is $\cap$-shaped with a maximum point.
• The graph crosses the $y$-axis at $(0,c)$, found by putting $x=0$. The turning point is called the vertex, and the vertical line through it is the line of symmetry. Every point on one side has a mirror image at the same height on the other side.

The roots of $f(x)=0$ are the $x$-coordinates where the graph meets the $x$-axis. Find them by factorising (or by the formula when the quadratic does not factorise). Example: $x^2-6x+5=(x-1)(x-5)$, so the roots are $1$ and $5$. If the roots are $p$ and $q$ the function can be written in factorised form $f(x)=a(x-p)(x-q)$, where $a$ is the coefficient of $x^2$. This lets you build a function from its graph. Example: roots $-1$ and $\frac52$ and leading coefficient $2$ give $f(x)=2(x+1)\left(x-\frac52\right)=2x^2-3x-5$. The graph can meet the $x$-axis twice, touch it once (a repeated root) or miss it completely.

The line of symmetry is halfway between the roots, and for any quadratic it is $x=-\frac{b}{2a}.$ Substitute this value into $f(x)$ to find the $y$-coordinate of the vertex. Example: for $y=-2x^2+8x-3$, $x=-\frac{8}{2(-2)}=2$ and $y=-8+16-3=5$, so the maximum is $5$ at $(2,5)$. Completing the square gives the vertex directly: $f(x)=a(x+p)^2+q$ has its vertex at $(-p,q)$. For example $x^2-6x+5=(x-3)^2-4$ has its minimum point at $(3,-4)$. The range is then $f(x)\ge q$ if $a>0$ or $f(x)\le q$ if $a<0$.

A sketch needs the right shape and the key coordinates, not accurate scale. Follow these steps.

1. Shape: $\cup$ if $a>0$, $\cap$ if $a<0$.
2. $y$-intercept: $(0,c)$.
3. Roots (the $x$-intercepts) by factorising or the formula.
4. Vertex, from symmetry or by completing the square. For $y=x^2-6x+5$: a $\cup$-shape through $(0,5)$, $(1,0)$ and $(5,0)$ with minimum point $(3,-4)$. Label each point with its coordinates, and make the curve symmetrical about $x=3$.

Points at the same height are the same distance either side of the line of symmetry. This lets you find a second point quickly. Example: for $y=-2x^2+8x-3$ the line of symmetry is $x=2$ and the $y$-intercept is $(0,-3)$. The mirror image is $(4,-3)$, so the line $y=-3$ meets the curve at $x=0$ and $x=4$. Check algebraically: $-2x^2+8x=0$ gives $x=0$ or $x=4$. A horizontal line $y=k$ meets a $\cap$-shaped graph twice when $k$ is below the maximum, once at the maximum and never above it. For a $\cup$-shaped graph reverse this about the minimum.

Quadratic functions model projectiles, areas and profit. The vertex gives the maximum or minimum value and the roots give when the quantity is zero. Example: a stone's height is $h=-5t^2+20t+25$. At $t=0$, $h=25$. The vertex is at $t=2$, $h=45$, so the maximum height is $45$ m. It reaches the ground when $h=0$: $-5(t-5)(t+1)=0$, so $t=5$ (reject $t=-1$). To find when $h$ exceeds $40$, solve $h=40$ to get $t=1$ and $t=3$. Because the graph is $\cap$-shaped, $h>40$ between these values, a duration of $2$ s. Always state the units and reject solutions outside the domain of the model.