C2.10 Graphs of functions

Learning objectives

C2.10.1Construct tables of values, and draw, recognise and interpret graphs• Construct tables of values, and draw, recognise and interpret graphs for functions of the following forms: • ax + b • ± x2 + ax + b • x a (x ≠ 0) where a and b are integer constants.C2.10.2Solve associated equations graphically, including finding and• Solve associated equations graphically, including finding and interpreting roots by graphical methods. e.g. find the intersection of a line and a curve.

A table of values turns a function rule into coordinates. Substitute each chosen $x$ -value to calculate $y$ , plot each $(x,y)$ pair on uniform scales, then join the points in the way the function requires.

Function form	Graph to recognise	Construction boundary
$y=ax+b$	a straight line	two accurate points determine the line; extra points check it
$y=\pm x^2+ax+b$	a smooth parabola	$+x^2$ opens upwards and $-x^2$ opens downwards
$y=\dfrac{a}{x}$ , $x\ne0$	two separate reciprocal branches	calculate values on both sides of zero and never join across $x=0$

Use this order: (1) copy the required $x$ -values; (2) substitute with brackets, especially for negative $x$ ; (3) check the table for arithmetic errors; (4) label axes and choose scales that cover every coordinate; (5) plot small accurate crosses; (6) draw a ruled straight line or a single smooth curve through the plotted pattern.

For $y=x^2+2x-4$ , the values at $x=-2,-1,0,1$ are $-4,-5,-4,-1$ . For instance, at $x=-2$ , $y=(-2)^2+2(-2)-4=-4$ . These coordinates form part of a smooth upward-opening parabola.

Interpret a graph by reading coordinates and visible features. The $y$ -intercept is where $x=0$ ; an $x$ -intercept is where $y=0$ . Read values from the stated scale and interpolate carefully between grid lines.

Do not connect reciprocal branches through $x=0$ : $a/x$ is undefined there. Do not replace a smooth quadratic or reciprocal curve with straight segments between plotted points.

A graphical solution is an $x$ -value where the required graphs have the same $y$ -value. The equation decides which intersection to read.

Equation	What to find on the graph
$f(x)=0$	where $y=f(x)$ crosses or touches the $x$ -axis
$f(x)=k$	intersections of $y=f(x)$ with the horizontal line $y=k$
$f(x)=g(x)$	intersections of the two graphs $y=f(x)$ and $y=g(x)$

Draw any additional line requested, identify every relevant intersection, then project vertically to the $x$ -axis. Record all $x$ -values allowed by the displayed domain. If coordinates are requested, read both $x$ and $y$ from each intersection.

To solve $x^2+2x-4=2x+2$ graphically, use the parabola $y=x^2+2x-4$ and the line $y=2x+2$ . Their intersections have equal $y$ -values, so the corresponding $x$ -coordinates are the solutions of the equation.

Graphical answers are usually approximate. Use the finest grid interval to estimate between scale marks, give sensible precision, and check that each reported point lies on both graphs. The number of intersections is the number of graphical solutions visible in the stated domain.

A root of $f(x)=0$ is an $x$ -coordinate, not the full coordinate pair. For an intersection equation, do not read where either graph crosses an axis unless that point is also an intersection of the required graphs.

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—C2.11 Sketching curvesRead

C2.10 Graphs of functions

Learning objectives

Construct and recognise linear, quadratic and reciprocal graphs

Solve equations by reading roots and intersections

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