Linear Graphs Flashcards

All 16 cards in this deck

What is the gradient of a line with equation $y = mx + c$?

e.g. $y = 2x + 3$

$m$, the coefficient of $x$.

So $y = 2x + 3$ has gradient $2$.

What are the coordinates of the point where the line $y = mx + c$ crosses the $y$-axis?

e.g. $y = -4x + 3$

$(0, c)$.

So $y = -4x + 3$ crosses at $(0, 3)$.

What are the steps to find the gradient of a line whose equation is not in the form $y = mx + c$?

e.g. $3y - 9x = 6$

1. Get the $y$ term alone: $3y = 9x + 6$
2. Divide every term by the number in front of $y$: $y = 3x + 2$
3. The gradient is the coefficient of $x$: $3$

What are the steps to find the equation of the line through two given points?

e.g. through $(0, 3)$ and $(2, -1)$

1. Gradient $m = \frac{\text{change in } y}{\text{change in } x} = \frac{-1-3}{2-0} = -2$
2. Find $c$ from a point on the line: the line crosses the $y$-axis at $(0,3)$, so $c = 3$
3. Write $y = mx + c$: $y = -2x + 3$

What are the steps to find the equation of a line through one point with a given gradient?

e.g. gradient $4$ through $(2, 9)$

1. Write $y = mx + c$ with the gradient: $y = 4x + c$
2. Substitute the point's coordinates: $9 = 4 \times 2 + c$
3. Solve for $c$ and write the equation: $c = 1$, so $y = 4x + 1$

How do you check algebraically whether a point lies on a given straight line?

e.g. does $(2, 9)$ lie on $y = 4x + 1$?

Substitute the $x$-coordinate into the equation and see if you get the $y$-coordinate.

$4 \times 2 + 1 = 9$, so yes, $(2, 9)$ lies on the line.

What are the steps to complete a table of values for a linear equation?

e.g. $y = 4x - 6$ for $x = -1, 0, 1$

1. Substitute the first $x$ value into the equation: $4 \times (-1) - 6 = -10$
2. Repeat for each $x$ value: $x = 0$ gives $-6$, $x = 1$ gives $-2$
3. Write each $y$ value under its $x$ value

What are the steps to draw the graph of $y = mx + c$ over a given range of $x$?

e.g. $y = \frac{1}{2}x - 1$ for $x$ from $-2$ to $3$

1. Complete a table of values: $y = -2, -1.5, -1, -0.5, 0, 0.5$
2. Plot the points, e.g. $(-2, -2)$ and $(3, 0.5)$
3. Join them with a single ruled straight line from $x = -2$ to $x = 3$

What does the graph of $x = -2$ look like?

A vertical line through $(-2, 0)$, parallel to the $y$-axis.

Every point on it has $x$-coordinate $-2$.

What does the graph of $y = -3$ look like?

A horizontal line through $(0, -3)$, parallel to the $x$-axis.

Every point on it has $y$-coordinate $-3$.

How do you find the gradient of a straight line drawn on a grid?

e.g. a line passing through $(1, 2)$ and $(4, 8)$

Gradient $= \frac{\text{change in } y}{\text{change in } x}$ between two points on the line.

$\frac{8-2}{4-1} = 2$

How can you tell from their equations that two straight lines are parallel?

They have the same gradient (the same value of $m$ in $y = mx + c$) but different values of $c$.

e.g. $y = 3x - 2$ and $y = 3x + 5$

What are the steps to show that two lines are parallel when one is not in the form $y = mx + c$?

e.g. $y = 3x - 2$ and $3y - 9x + 5 = 0$

1. Rearrange into $y = mx + c$: $3y = 9x - 5$, so $y = 3x - \frac{5}{3}$
2. Compare the coefficients of $x$: both are $3$
3. State the conclusion: both lines have gradient $3$, so they are parallel

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

Any line $y = 5x + c$ with $c \neq 0$.

e.g. $y = 5x + 2$

True or false? The lines $y = 2x + 1$ and $y = 3x + 1$ are parallel.

False.

Their gradients differ ($2$ and $3$); they both pass through $(0, 1)$, so they cross there.

True or false? Two different parallel lines can have the same $y$-intercept.

False.

Same gradient and same $y$-intercept means they are the same line.