Solving Problems On Linear Graphs

 

Use the graph below to answer the following questions 

1. What is the y-intercept of the straight line shown in the graph?

• A) (1, 0)
• B) (0, 1)
• C) (0, -1)
• D) (-0.5, 0)

Correct Answer: B) (0, 1)

Solution: The y-intercept is the point where the line crosses the vertical y-axis (where x = 0). Looking closely at the graph, the line crosses the y-axis exactly at 1. The graph explicitly labels this coordinate as (0, 1).

2. What is the slope (gradient) of the line represented in the graph?

• A) 1
• B) -2
• C) 2
• D) 1/2

Correct Answer: C) 2

Solution: To find the slope (m), pick any two points marked with an "X" on the graph. For example, using (0, 1) and (1, 3):
m = (y₂ - y₁) / (x₂ - x₁)
m = (3 - 1) / (1 - 0) = 2 / 1 = 2.

Alternatively, observe that for every 1 unit you move to the right along the x-axis, the line rises by 2 units along the y-axis (Rise/Run = 2/1 = 2).

3. Which of the following equations perfectly defines the straight line in the graph?

• A) y = x + 1
• B) y = 2x + 1
• C) y = 2x - 1
• D) y = -2x + 1

Correct Answer: B) y = 2x + 1

Solution: The slope-intercept form of a linear equation is given by: y = mx + c, where m is the slope and c is the y-intercept. From the previous questions, the slope m = 2 and the y-intercept c = 1. Substituting these values into the formula yields: y = 2x + 1.

4. Based on the line's trajectory and markings in the graph, at what value of x does the line cross the horizontal x-axis?

• A) x = -0.5
• B) x = -1
• C) x = 0
• D) x = 0.5

Correct Answer: A) x = -0.5

Solution: The x-intercept occurs where y = 0. Using the equation of the line (y = 2x + 1), set y to 0 and solve for x:
0 = 2x + 1
-1 = 2x
x = -1/2 = -0.5.

Visually checking the graph confirms the blue line crosses the horizontal axis exactly halfway between 0 and -1.

5. Which of the following ordered pairs is NOT a point marked with an orange "X" on the line in the graph?

• A) (2, 5)
• B) (-1, -1)
• C) (-3, -5)
• D) (-2, -4)

Correct Answer: D) (-2, -4)

Solution: Check the options using the line's equation y = 2x + 1:
• For x = 2: y = 2(2) + 1 = 5 → (2, 5) is marked.
• For x = -1: y = 2(-1) + 1 = -1 → (-1, -1) is marked.
• For x = -3: y = 2(-3) + 1 = -5 → (-3, -5) is marked.
• For x = -2: y = 2(-2) + 1 = -3 → The marked point on the line is (-2, -3). Therefore, (-2, -4) does not lie on the line.