Graphs

Spec reference: Algebra - Graphs
Key idea: Draw and interpret straight-line and curved graphs; find gradient and equation of a line.

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Straight-line graphs: $y=mx+c$

• $m$ = gradient (steepness)
• $c$ = y-intercept (where the line crosses the y-axis)

Example

Draw $y=2x+1$

$x$	$y=2x+1$
$-1$	$-1$
$0$	$1$
$2$	$5$

Plot the points and draw a straight line.

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Gradient

$$m=\frac{\text{rise}}{\text{run}}=\frac{y_2-y_1}{x_2-x_1}$$

Example

Find the gradient of the line through $(2,3)$ and $(6,11)$.

$$m=\frac{11-3}{6-2}=\frac{8}{4}=\boxed{{\displaystyle 2}}$$

• Positive gradient - line slopes upward left to right
• Negative gradient - line slopes downward left to right
• $m=0$ - horizontal line
• Vertical line has undefined gradient

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Finding the equation of a line

Given gradient and a point:

Use $y-y_1=m(x-x_1)$

Example

Find the equation of the line with gradient 3 passing through $(2,5)$.

$y-5=3(x-2)$

$y=3x-6+5=\boxed{{\displaystyle 3x-1}}$

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Parallel and perpendicular lines

• Parallel lines have the same gradient
• Perpendicular lines have gradients that multiply to give $-1$: $m_1\times m_2=-1$

Example

A line has gradient 2. What is the gradient of a perpendicular line?

$m=-{\displaystyle \frac{1}{2}}$

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Horizontal and vertical lines

• $y=a$ is a horizontal line (e.g. $y=3$)
• $x=a$ is a vertical line (e.g. $x=-2$)

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Quadratic graphs

$y=a{x}^2+bx+c$ produces a parabola (U-shape if $a>0$, n-shape if $a<0$).

Example

Draw $y=x^2-3x+2$

$x$	$-1$	$0$	$1$	$2$	$3$	$4$
$y$	$6$	$2$	$0$	$0$	$2$	$6$

The parabola has roots at $x=1$ and $x=2$.

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Real-life graphs

• Distance-time graphs: gradient = speed; horizontal line = stationary
• Speed-time graphs: gradient = acceleration; area under graph = distance

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Exam tips

Watch out for

• Always use at least 3 points to draw a straight-line graph
• A negative gradient does NOT mean a negative y-intercept
• The roots of a quadratic are where the graph crosses the x-axis ($y=0$)

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Test Yourself

Question 1 of 5

Gradient of $y = 3x - 7$?