Free NJGPA Interpreting Functions and Key Features from Graphs and Tables (Algebra I) Questions and Answers 1 — Questions and Answers
Question 1: A function is represented by a continuous graph that starts at the point (-4, 2) and ends at the point (5, -1). What is the domain of the function?
- [-1, 2]
- [-4, 5] (Correct answer)
- All real numbers
- {-4, 5}
Correct answer: [-4, 5]
The domain of a function refers to all possible x-values. Since the graph is continuous and extends from an x-value of -4 to an x-value of 5, inclusive, the domain is all real numbers between -4 and 5, which is written as [-4, 5] or -4 ≤ x ≤ 5.
Question 2: Consider the function f(x) represented by the table of values below. What is the y-intercept of the function? Table: x: -2, -1, 0, 1, 2; f(x): 9, 6, 3, 0, -3
- (1, 0)
- (0, 1)
- (3, 0)
- (0, 3) (Correct answer)
Correct answer: (0, 3)
The y-intercept is the point where the graph of the function crosses the y-axis. This always occurs when the x-value is 0. According to the table, when x = 0, the value of f(x) is 3, so the y-intercept is (0, 3).
Question 3: The graph of a parabola opens downwards and has its vertex at the point (2, 8). On which interval is the function increasing?
- (-∞, 2) (Correct answer)
- (2, ∞)
- (-∞, 8)
- (8, ∞)
Correct answer: (-∞, 2)
Since the parabola opens downwards, the function's values increase as x approaches the vertex from the left and decrease as x moves away from the vertex to the right. The x-coordinate of the vertex is 2. Therefore, the function is increasing on the interval from negative infinity up to 2, which is written as (-∞, 2).
Question 4: The graph shows the height of a ball thrown into the air over time. The highest point on the graph is (3, 25). What is the maximum height the ball reaches?
- 3
- 25 (Correct answer)
- 28
- It cannot be determined from the graph.
Correct answer: 25
The maximum value of a function shown on a graph corresponds to the y-coordinate of the highest point (the vertex or a peak). In this case, the point (3, 25) means that at 3 seconds (x-value), the height is 25 feet (y-value). Therefore, the maximum height is 25.
Question 5: A function is defined by the following table of values. What is the average rate of change over the interval from x = 1 to x = 4? Table: x: 0, 1, 2, 3, 4; f(x): 2, 5, 10, 17, 26
Correct answer: 7
The average rate of change is calculated as the change in y divided by the change in x, or (f(b) - f(a)) / (b - a). Here, a=1 and b=4. From the table, f(4) = 26 and f(1) = 5. So, the calculation is (26 - 5) / (4 - 1) = 21 / 3 = 7.
Question 6: The graph of a function d(t) shows the distance (in miles) a car has traveled from its starting point after t hours. What does the point (3, 150) on the graph represent?
- The car traveled 3 miles in 150 hours.
- The car's average speed is 150 miles per hour.
- After 3 hours, the car has traveled 150 miles. (Correct answer)
- The car will travel a total of 150 miles on its trip.
Correct answer: After 3 hours, the car has traveled 150 miles.
In a function, the point (x, y) relates the input (x) to the output (y). In this context, t is the input and d(t) is the output. Therefore, the point (3, 150) means that when the time (t) is 3 hours, the distance traveled (d) is 150 miles.
Question 7: Function A is represented by the equation y = 4x + 5. Function B is represented by a graph that is a straight line passing through the points (0, 3) and (2, 7). Which statement is true?
- The y-intercept of Function A is less than the y-intercept of Function B.
- The y-intercept of Function A is greater than the y-intercept of Function B. (Correct answer)
- The y-intercepts of both functions are equal.
- The rate of change of Function B is greater than the rate of change of Function A.
Correct answer: The y-intercept of Function A is greater than the y-intercept of Function B.
The y-intercept is the value of y when x=0. For Function A, the equation is in slope-intercept form (y=mx+b), so the y-intercept (b) is 5. For Function B, the graph passes through (0, 3), which is its y-intercept. Since 5 > 3, the y-intercept of Function A is greater than the y-intercept of Function B.
