Function Analysis — Basic Graph Reading (Part B, Variant 2)
Function Analysis — Basic Graph Reading (Part B, Variant 2). Practice questions to deepen understanding of basic graph reading in function analysis — additional practice. Online math practice with full solutions and step-by-step explanations.
Basic graph reading practice — finding f(x) and x from the graph, identifying increase/decrease, matching a table to a graph, and locating maximum and minimum points. 70 interactive questions.
Which line has the steepest slope?
Solution: A steep slope = rapid y change. Line C rises the fastest.
From the graph, find the value of \(f(2)\)
Solution: At x=2 the graph is at height y=3.
Find the value of \(f(1)\) from the graph:
Solution: The graph represents \(y=x\), so f(1)=1.
What is \(f(2)\) according to the graph?
Solution: The graph is a horizontal line at \(y=3\), so f(2)=3.
What is the value of \(f(0)\) according to the graph?
Solution: The graph passes through the origin: (0,0).
Calculate \(f(3)\) from the following graph:
Solution: The point above x=3 is at height y=4.
Calculate \(f(-2)\) from the graph:
Solution: At \(x=-2\) the graph gives y=−2.
What is \(f(1)\) according to the graph?
Solution: The point on the graph where x=1 is at height y=2.
From the graph, what is \(f(-1)\)?
Solution: On the graph y=−x, when x=−1: y=1.
Find \(f(1.5)\) from the graph:
Solution: The point at x=1.5 is at height y=4.
What is \(f(2)\) from the graph?
Solution: The point at x=2 is at height y=4.
From the graph, find x when \(f(x)=2\):
Solution: Draw a horizontal line at y=2; it meets the graph at x=1.
Find x when \(f(x)=0\):
Solution: f(x)=0 means the x-intercept.
For which x does \(f(x)=4\)?
Solution: The horizontal line y=4 meets the graph at x=2.
For which x does \(f(x)=-1\)?
Solution: The horizontal line at y=−1 meets the graph at x=−1.
Find x when \(f(x)=3\):
Solution: The line y=3 crosses the graph at x=2.
At which x does the function equal 0?
Solution: The graph crosses the x-axis at the origin.
Find x when \(f(x)=2\):
Solution: The horizontal line y=2 meets the graph at x=1.5.
How many x values satisfy \(f(x)=2\)?
The horizontal line \(y=2\) intersects the graph at three points.
Therefore there are 3 different values of x for which \(f(x)=2\).
Answer: 3 values.
For which x does \(f(x)=5\)?
Solution: The line y=5 meets the graph at x=3.
In which interval is the function increasing?
Solution: The graph rises throughout → increasing everywhere.
In which interval is the function decreasing?
Solution: The graph falls throughout → decreasing everywhere.
In which interval is the function increasing?
Solution: The graph decreases from 0 to 1.5, then increases.
Is the function increasing or decreasing?
Solution: A horizontal graph = constant function.
In which interval is the function decreasing?
Solution: An inverted parabola: rises to the vertex, then falls.
In which intervals is the function increasing?
Solution: The graph rises on [0,1], then on [1.5,3].
At which point does the function stop increasing and start decreasing?
Solution: The point where the function changes from increasing to decreasing is called the maximum.
Is the function increasing on [1,2]?
Solution: On [1,2] the graph moves upward.
In which interval does the function decrease the fastest?
Solution: A steep graph → rapid change. The second interval is steeper.
How many increasing intervals does the function have?
Solution: The graph rises, then falls, then rises again, then falls → 3 increasing segments.
At which point does the function have a maximum?
Solution: Maximum = the highest point on the graph.
At which point does the function have a minimum?
Solution: Minimum = the lowest point on the graph.
What is the maximum value of the function?
Solution: To find the maximum, look for the highest point on the graph.
How many minimum points does the function have?
Solution: Count the local lowest points on the graph.
Does the function have a maximum? (The function continues without bound)
Solution: The graph rises indefinitely → no maximum.
What is the minimum value of \(f(x)=x^2+1\)?
Solution: This is a parabola f(x)=x²+1. Minimum at x=0: y=1.
At which x does the function reach its peak?
Solution: The peak is the maximum point; the graph rises to x=2 then falls.
What is the range of \(f(x)=-x^2+4\)?
Solution: An inverted parabola with vertex at y=4 → range: all y ≤ 4.
How many extreme points does the function have?
Solution: Count all local maxima and minima on the graph.
At which point is the function at its minimum?
Solution: An upright parabola has its minimum (vertex) at x=0.