4 Graphs

Learning objectives

44A Graphical and numerical formsTranslate information between graphical and numerical form.44B Linear relationshipsUnderstand that y = mx + c represents a linear relationship.44C Plot two variablesPlot two variables, discrete or continuous, from experimental or other data.44D Slope and interceptDetermine the slope and intercept of a linear graph.44E Tangents and rate of changeUnderstand, draw and use a curve tangent's slope as a measure of rate of change.

A graph and a table can represent the same relationship: each plotted point pairs one value on the horizontal axis with one value on the vertical axis.

Task	Construction	Numerical result
find $y$ for a given $x$	move vertically from $x$ to the graph, then horizontally to the $y$ -axis	read $y$ with its unit
find $x$ for a given $y$	move horizontally from $y$ to the graph, then vertically to the $x$ -axis	read $x$ with its unit
make a table	record the two coordinates of selected points	one row is one $(x,y)$ pair

Read each axis label, unit and scale before estimating. Interpolation estimates between plotted values. An intersection gives the pair of values shared by two graphs; a plateau shows that the vertical value is no longer changing as the horizontal value increases.

Keep construction lines parallel to the axes and report a precision justified by the scale. Extrapolation beyond the measured range is less secure than interpolation and should not be treated as an observed value.

A relationship of the form $y=mx+c$ is linear because equal changes in $x$ produce equal changes in $y$ ; its graph is a straight line.

y=mx+c

Term	Meaning on a graph of $y$ against $x$
$m$	gradient: the change in $y$ for each unit change in $x$
$c$	vertical-axis intercept: the value of $y$ when $x=0$
$m>0$	the line rises from left to right
$m<0$	the line falls from left to right
$m=0$	$y$ is constant, so the line is horizontal

For $y=3x+2$ , increasing $x$ by $1$ increases $y$ by $3$ , and the line crosses the $y$ -axis at $2$ . The points $(0,2)$ , $(1,5)$ and $(2,8)$ therefore lie on the same straight line.

A straight-line relationship is directly proportional only when $c=0$ , so the line passes through the origin. A straight line with a non-zero intercept is linear but not directly proportional.

Plotting two variables turns paired data into a graph so that a pattern, relationship or anomalous result can be seen.

Step	Action
1	place the independent variable on the horizontal axis and the dependent variable on the vertical axis
2	label both axes with the variable and unit
3	choose simple, even scales that use most of the grid
4	plot every coordinate accurately as a small cross
5	draw a suitable line or smooth curve of best fit through the overall trend

Continuous measurements, such as temperature and time, may take values between recorded points and usually support a line or curve. Discrete values occur as separate counts or categories and should remain distinct. A best-fit construction should balance the scatter rather than join each point in turn.

Do not force an anomalous point into the trend or omit it without evidence. Keep it plotted, identify it as a possible anomaly, and use the pattern of the remaining data when judging the best fit.

The gradient measures how quickly $y$ changes with $x$ , while the intercept is the value of $y$ where the line crosses the vertical axis.

m=rac{\Delta y}{\Delta x}=rac{y_2-y_1}{x_2-x_1},\qquad c=y-mx

Choose two widely separated points on the straight best-fit line, not necessarily measured data points. Form a large gradient triangle, calculate vertical change divided by horizontal change, then substitute one point into $c=y-mx$ or read the intercept at $x=0$ .

ext{using }(2,5) ext{ and }(8,17):\quad m=rac{17-5}{8-2}=2,\quad c=5-(2 imes2)=1,\quad y=2x+1

Gradient units are $y$ -axis units divided by $x$ -axis units. Preserve the sign: a falling line has a negative gradient. A small triangle magnifies reading error, so use much of the line where possible.

For a curved graph, the gradient changes from point to point. The gradient of a tangent at one point measures the instantaneous rate of change there.

Step	Action
1	mark the point at which the rate is required
2	draw a straight tangent that follows the curve's local direction and touches it at that point
3	choose two widely separated points on the tangent
4	calculate $\Delta y/\Delta x$ and state the derived rate unit

ext{instantaneous rate}= ext{tangent gradient}=rac{\Delta y}{\Delta x}

On a hydrogen-volume against time graph, the tangent gradient has units such as $\mathrm{cm^3\,s^{-1}}$ . A steep tangent means a faster rate; as the curve becomes less steep, the instantaneous rate decreases.

A chord joining two points on the curve gives an average rate over an interval, not the rate at one instant. Calculate the gradient from points on the tangent, not from nearby points on the curve.

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—Pearson Edexcel International GCSE Chemistry (4CH1)Read

4 Graphs

Learning objectives

Translate between graphs and numerical values

Recognise a linear relationship

Plot two variables from data

Determine the gradient and intercept of a line

Use a tangent to find an instantaneous rate

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