MathLab with Mr. Ahmed is a math learning website by Ahmed Amasha, a mathematics teacher teaching the American curriculum. Explore Algebra 2, Precalculus, Calculus, and SAT and PSAT Math Prep through lessons, focused practice and printable resources.
2.3 Getting Information from the Graph of a Function
An x-intercept has y=0. A y-intercept has x=0. Increasing means outputs rise as inputs increase.
Remember State both the input and output when reporting an extremum.
Worked example 1
01 / Read f(−1),f(1), and the net change from x=−1 to x=1 from the graph.
02 / At x=−1, the graph height is 0.
03 / At x=1, the height is 4.
04 / Final minus initial is ((2)+2)−((2)−2)=4.
Worked example 2 · Read graph values, level sets, and net change
From this complete graph, find every input with f(x)=3.
Draw the horizontal level y=3. It meets both included endpoints and the middle rising segment. The three input coordinates are −3,0,3.
Read the graphs
Read graph values, level sets, and net change
Read f(−1),f(1), and the net change from x=−1 to x=1 from the graph.
At x=−1, the graph height is 0. At x=1, the height is 4. Final minus initial is ((2)+2)−((2)−2)=4.
Answer: 0,4, net change 4.
Read domains and ranges with holes and endpoints
Find the domain and range of the complete graph.
The left input endpoint is excluded and the right endpoint included. The interior low point at x=0 is included. The highest attained height is at the right endpoint.
Answer: D=(−2,3],R=[0,3]
Compare graphs and locate intersections
Write your comparisons before selecting the intersection answer. Also compare f(0) with g(0), and f(3) with g(3). The curves continue beyond the viewing window. Find the solutions of f(x)=g(x), and distinguish the solutions from intersection points. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
Look for equal heights, not x-intercepts. The two intersections occur symmetrically at x=±(2). The equation asks for the x-coordinates. At input 0,f(0)=0<4=g(0). At input 3, the parabola has height 9>4. Written-work review:
Correct comparison at both specified inputs.
Both intersections located.
Input solutions distinguished from coordinate pairs.
Answer: x=−2 or x=2; points (−2,4),(2,4).
Solve equations and inequalities graphically
Solve H(x)=0 and H(x)≥0 using the graph. The formula is H(x)=(x−2)(x−1)(x+2) and the curve continues beyond the window.
Read all three crossings of the horizontal axis. The graph alternates signs across simple zeros. Its far-right sign is positive and equality includes each zero.
Answer: Zeros: −2,1,2;H≥0 on [−2,1]∪[2,∞)
Identify increasing, decreasing, and constant intervals
State the intervals where the graphed function is increasing, decreasing, or constant. Use open intervals for monotonic behavior.
Read the curve from smaller x to larger x. Positive slopes indicate increasing behavior, a horizontal piece is constant, and a negative slope is decreasing. A constant piece is not strictly increasing.
Answer: Increasing (0,2) and (6,8); decreasing (4,6); constant (2,4).
Combine formula and graph information
Draw your own full graph using the formula, label the key features, then analyze it. For f(x)=x2−2, use the graph and formula to state full domain, range, and decreasing/increasing intervals. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
The polynomial has all real inputs, beyond the displayed window. The squared term has its minimum zero at x=(0). The graph falls toward the vertex and rises away from it. Written-work review:
Correct graph with vertex or turning points.
Full domain and range distinguished from the viewing rectangle.
All requested monotonic intervals identified.
Answer: D=R,R=[−2,∞); decreases on (−∞,0), increases on (0,∞).
Distinguish local extrema, coordinates, and global bounds
Use the entire graph to list interior local extrema as “value at input,” then give the absolute maximum and minimum. Endpoints are excluded from the term interior local extremum.
Compare each turn with nearby graph heights. An extremum value is the vertical coordinate, not its input. For absolute extrema, compare both interior turns and included endpoints.
Answer: Local maxima 4 at −2 and 3 at 2; local minimum 2 at 0. Absolute maximum 4 at −2; absolute minimum −1 at −4.
Approximate polynomial turning points using technology
Use graphing technology to find every local extremum of f(x)=x3−2x to two decimals, and state increasing/decreasing intervals. Write or draw your full response before checking the answer. A teacher must review the construction or argument. Also state the full domain and range, not merely the displayed window. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
Locate both turns in a broad window and zoom each separately. Refine their input locations and evaluate the function there. The left turn is a maximum and the right turn a minimum; the cubic rises on the two exterior intervals. The polynomial domain is all real numbers. Its odd degree and nonzero leading coefficient give range all real numbers. Written-work review:
Both turning points refined to the requested precision.
Values and locations not reversed.
All three monotonic intervals given with rounded boundaries identified as approximate.
Answer: Maximum 1.09 at x=−0.82; minimum −1.09 at x=0.82; increasing outside those inputs and decreasing between them.
Interpret graphical stories in context
Read the modeled power consumption at 6h and 18h, the time of minimum and maximum, and the net change from 6h to 18h.
Read heights at the two specified times. The lowest vertex is at 3h and the highest at 12h. Subtract later minus earlier power; the result has power units, not hours.
Answer: 6MW and 8MW; minimum at 3h, maximum at 12h; net change +2MW.
Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.