Linear Graphs Flashcards

All 23 cards in this deck

In the equation $y = mx + c$, what do $m$ and $c$ stand for?

$m$ is the gradient and $c$ is the $y$-intercept (where the line crosses the $y$-axis).

What is the formula for the gradient of the line through two points?
e.g. $(1, 2)$ and $(4, 8)$

$m = \dfrac{y_2 - y_1}{x_2 - x_1}$
e.g. $\dfrac{8-2}{4-1} = 2$

What are the steps to rearrange an equation into the form $y = mx + c$ and state the gradient and intercept?
e.g. $3y - 9x + 5 = 0$

1. Get the $y$ term alone: $3y = 9x - 5$
2. Divide every term by the coefficient of $y$: $y = 3x - \frac{5}{3}$
3. Read off: gradient $3$, $y$-intercept $-\frac{5}{3}$

What are the steps to find the equation of the line through two given points?
e.g. $(1, 5)$ and $(3, 11)$

1. Gradient $= \frac{11-5}{3-1} = 3$
2. Substitute one point into $y = 3x + c$: $5 = 3 + c$, so $c = 2$
3. Equation: $y = 3x + 2$

What are the steps to find the equation of a line with a given gradient through a given point?
e.g. gradient $4$ through $(2, 5)$

1. Write $y = 4x + c$
2. Substitute the point: $5 = 4 \times 2 + c$, so $c = -3$
3. Equation: $y = 4x - 3$ (or $y - 5 = 4(x - 2)$)

What are the steps to draw the graph of a straight line from its equation?
e.g. $y = 2x - 1$ for $x$ from $-1$ to $3$

1. Make a table of values: $x = -1, 0, 1, 2, 3$
2. Work out $y$ for each: $-3, -1, 1, 3, 5$
3. Plot the points and join them with a single straight line, extended over the given range

What is the smallest number of points you should plot when drawing a straight-line graph, and why?

Three — two fix the line and the third checks the points are in line.

What are the steps to find the equation of a line drawn on a grid?
e.g. a line through $(0, 3)$ and $(2, -1)$

1. Gradient from two points on the line: $\frac{-1-3}{2-0} = -2$
2. Read the $y$-intercept: $c = 3$
3. Write $y = mx + c$: $y = -2x + 3$

How do you find where a straight line crosses each axis from its equation?
e.g. $y = 2x - 6$

Put $x = 0$ for the $y$-axis crossing and $y = 0$ for the $x$-axis crossing.
e.g. $(0, -6)$ and $(3, 0)$

What does the graph of $x = 3$ look like?

A vertical line through $(3, 0)$, parallel to the $y$-axis.

True or false? The graph of $y = 4$ is a vertical line.

False. It is a horizontal line through $(0, 4)$; $x = 4$ is the vertical one.

What is true about the gradients of two parallel lines?

They are equal.

What are the steps to show that two lines are parallel?
e.g. $y = 2x + 3$ and $5y - 10x + 4 = 0$

1. Rearrange each into $y = mx + c$: $y = 2x - \frac{4}{5}$
2. Compare the gradients: both are $2$
3. State: same gradient, so the lines are parallel

Write down the equation of a straight line parallel to $y = 5x$.

$y = 5x + c$ for any $c \neq 0$, e.g. $y = 5x + 2$.

True or false? Two lines that cross the $y$-axis at the same point are parallel.

False. Parallel lines need equal gradients; lines sharing a $y$-intercept cross there unless they are the same line.

True or false? $y = 3x + 1$ and $y = 3x - 7$ are parallel.

True. Both have gradient $3$ and different $y$-intercepts.

True or false? You can compare the gradients of $y = \frac{1}{2}x - 6$ and $6y = 3x + 7$ straight away, without rearranging.

False. The second must be written as $y = mx + c$ first, giving $y = \frac{1}{2}x + \frac{7}{6}$.

What is the product of the gradients of two perpendicular lines?

$-1$, i.e. $m_1 m_2 = -1$.

What is the gradient of a line perpendicular to a line of gradient $m$?
e.g. $m = \frac{3}{2}$

The negative reciprocal, $-\frac{1}{m}$.
e.g. $-\frac{2}{3}$

What are the steps to find the equation of the line perpendicular to a given line through a given point?
e.g. perpendicular to $y = 3x - 4$ through $(9, 5)$

1. Negative reciprocal of the gradient: $-\frac{1}{3}$
2. Substitute the point into $y = -\frac{1}{3}x + c$: $5 = -3 + c$, so $c = 8$
3. Equation: $y = -\frac{1}{3}x + 8$

True or false? A line perpendicular to $y = 2x + 1$ has gradient $-2$.

False. The gradient is $-\frac{1}{2}$, the negative reciprocal of $2$.

True or false? $y = -\frac{1}{2}x + 3$ and $y = 2x - 1$ are perpendicular.

True. $-\frac{1}{2} \times 2 = -1$.

What is the equation of a line perpendicular to the horizontal line $y = 4$?

A vertical line $x = a$ for some number $a$, e.g. $x = 2$.