Unit 4 Linear Functions Test Study Guide - WordPress.com

IB Math Studies Yr 1 Final Exam Review

Name_________________________________ Date: ____________

For this quiz, you need to be able to:

1. Find the gradient, distance, and midpoint of 2 points on a line.

2. Identify the gradient and y-intercept of a line from a graph or equation.

3. Write the equation of a line in gradient-intercept form (𝑦 = 𝑚𝑥 + 𝑐)

4. Write the equation of a line in standard form (𝑎𝑥 + 𝑏𝑦 + 𝑑 = 0)

5. With gradient, you need to be able to find parallel and perpendicular gradients.

PART 1: Find the gradient, distance, and midpoint of 2 points on a line.

Distance formula

The distance formula is used to find the length between any 2 points.

The distance (or length) of a line is calculated by using the formula below from the “Prior learning”

section (page 2) of the IB Formula Booklet.

Example: Find the length between (4, -5) and (-1, -1).

Unit 4 Linear Functions Test Study Guide


Midpoint formula

The midpoint formula is used to find the “middle point” between any 2 points.

The midpoint of any 2 points can be found by using the formula below from the “Prior learning” section

(page 2) of the IB Formula Booklet.

Example: Find the midpoint between (-9, -1) and (-3, 7).

Gradient formula (slope)

The gradient formula (slope) is used to calculate the direction and steepness between 2 points on a line.

The gradient of any 2 points can be found by using the formula below from the “Topic 5 – Geometry

and trigonometry” section (page 5) of the IB Formula Booklet.

Example: Find the gradient between (-1, 3) and (5, 1).


PART 2: Identify the gradient and y-intercept of a line from a graph or equation.

Equation

Find the gradient and y-intercept of 4x + 3y = 9.

Graph

Find the gradient and y-intercept of the graph

below.

𝑚 =𝑟𝑖𝑠𝑒

𝑟𝑢𝑛=

1

2 𝑐 = 1

PART 3: Write the equation of a line in gradient-intercept form (𝑦 = 𝑚𝑥 + 𝑐)

The equation of a line in slope-intercept form tells us two key pieces of information.

Here are two examples of how to write the equation of a line given the gradient and a point.


Here is an example of how to write the equation of a line given 2 points.


PART 4: Write the equation of a line in standard form (𝑎𝑥 + 𝑏𝑦 + 𝑑 = 0)

When you are asked to write an equation in standard form, it is asked as:

“Please write the following equation in the form 𝒂𝒙 + 𝒃𝒚 + 𝒅 = 𝟎 , where a > 0, and 𝒂, 𝒃, 𝒂𝒏𝒅 𝒅 ∈ ℤ.”

When you see these directions:

1. Write the equation in gradient-intercept form.

2. Get everything to one side of the equation (Set equation = 0)

3. Make sure that a or the coefficient with a is positive

4. Multiply by the denominator of the fraction in order to make a, b, and c integers.

Here is an example of how to write the equation of a line in standard form when given 2 points.

A line passes through the points (-2, 6) and (1, 4). Write down the equation of in the form 𝑎𝑥 + 𝑏𝑦 + 𝑑 = 0, where 𝑎, 𝑏, and 𝑑 are integers and 𝑎 > 0.

First write the equation in gradient-intercept form (𝑦 = 𝑚𝑥 + 𝑐)

𝑚 =𝑦2 − 𝑦1

𝑥2 − 𝑥1=

4 − 6

1 − (−2)=

−2

3= −

2

3

𝑦 = 𝑚𝑥 + 𝑐

𝑦 = −2

3𝑥 + 𝑐

Use one of the given points to find “c”. Let’s use (1, 4)

4 = −2

3(1) + 𝑐

4 = −2

3+ 𝑐

𝑐 =14

3

𝑦 = −2

3𝑥 +

14

3

Now convert to standard form (𝑎𝑥 + 𝑏𝑦 + 𝑑 = 0)

𝑦 = −2

3𝑥 +

14

3

0 = −2

3𝑥 − 𝑦 +

14

3 (divide all by -1)

0 =2

3𝑥 + 𝑦 −

14

3 (multiple all by 3)

0 = 2𝑥 + 3𝑦 − 14


PART 5: With gradient, you need to be able to find parallel and perpendicular gradient.

Two lines are parallel if and only if they have the same gradient.

Same gradient means equal slopes

Two lines are perpendicular if and only if their equations form a 900 angle at their point of intersection.

We can also determine whether two lines are perpendicular if their gradients

are negative reciprocals of each other

Negative reciprocal means to flip and negate a fraction.

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Unit 4 Linear Functions Test Study Guide - WordPress.com