Patterns And Relations (9)

Question 1

A pattern starts with 2, 5, 8, 11… What type of pattern is this and why?

Answer 1

This is an arithmetic pattern; each term increases by a common difference of 3

Question 2

Write a rule for the pattern 3, 6, 12, 24… Explain your reasoning.

Answer 2

Each term multiplies by 2; rule: next term = previous term × 2

Question 3

How can you represent the pattern 4, 7, 10, 13 in a table?

Answer 3

Column 1: term number (1,2,3,4), Column 2: term value (4,7,10,13); shows relationship between term number and value

Question 4

How can you represent the same pattern with a graph?

Answer 4

Plot term number on x-axis and term value on y-axis; points lie on a straight line since it is arithmetic

Question 5

A line passes through points (1,3) and (4,9). How do you find the slope?

Answer 5

Slope m = (y2 - y1)/(x2 - x1) = (9 - 3)/(4 - 1) = 6/3 = 2

Question 6

Using the previous line, write the slope-intercept form.

Answer 6

y = mx + b; m = 2, use point (1,3): 3 = 2*1 + b → b = 1; Equation: y = 2x + 1

Question 7

What does the slope tell you about a line?

Answer 7

Slope shows the rate of change; for every 1 unit increase in x, y increases by the slope value

Question 8

What does the y-intercept tell you in a real-world context?

Answer 8

Y-intercept is the starting value of y when x = 0

Question 9

Solve 3x + 5 = 20 and explain each step.

Answer 9

Subtract 5: 3x = 15, divide by 3: x = 5; isolate variable to solve linear equation

Question 10

A pattern is described by y = 4x - 2. What is the y-value when x = 5?

Answer 10

y = 4*5 - 2 = 20 - 2 = 18; plug in x into the equation

Question 11

A taxi charges a $3 base fee plus $2 per kilometer. Write an equation for the total cost y based on kilometers x.

Answer 11

y = 2x + 3; slope = 2 (cost per km), y-intercept = 3 (base fee)

Question 12

Using the taxi equation y = 2x + 3, what is the cost for 10 km?

Answer 12

y = 2*10 + 3 = 23; plug in x = 10 into the equation

Question 13

A line passes through points (2,5) and (5,11). Which slope formula do you use and why?

Answer 13

m = (y2 - y1)/(x2 - x1); calculates rate of change between two points on a line

Question 14

The graph of a linear relation shows a straight line increasing from left to right. What does this indicate about the slope?

Answer 14

The slope is positive; y increases as x increases

Question 15

A graph shows a horizontal line at y = 7. What does the slope and y-intercept tell you?

Answer 15

Slope = 0 (no change in y), y-intercept = 7; y is constant regardless of x

Question 16

A phone plan charges a flat $10 per month plus $0.50 per text message. Which formula represents this situation and explain the variables.

Answer 16

y = 0.5x + 10; x = number of texts, y = total cost, slope = cost per text, y-intercept = flat monthly fee

Question 17

Given two equations y = 3x + 2 and y = 2x + 5, which line is steeper and why?

Answer 17

Compare slopes: 3 > 2; y = 3x + 2 is steeper because slope represents rate of change

Question 18

Solve for x: 5x - 7 = 18. Show each step.

Answer 18

Add 7: 5x = 25; divide by 5: x = 5

Question 19

The pattern of weekly savings is $5, $10, $15, $20… Write an equation for total savings y after x weeks.

Answer 19

y = 5x; slope = 5 (savings per week), y-intercept = 0 (starting with $0)

Question 20

You are given a table of values for a linear relation: x: 0,1,2,3; y: 4,7,10,13. How do you find the equation of the line?

Answer 20

Find slope: (7-4)/(1-0) = 3; y-intercept = 4; Equation: y = 3x + 4