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Grade 11 Precalculus · Lesson 2.3

Getting Information from the Graph

Read a graph as mathematical evidence: scan its axes, project points, identify intervals, compare functions, and justify every conclusion.
9 learning scenesInteractiveGraph-first learning
Explanation

01 · Explore

Read the Graph, Not Just the Equation

Move the scanner along the graph. Read the input on the horizontal axis and the output on the vertical axis. A missing point is not in the domain.

Explanation

Mr. Azzam's method · Read a coordinate

  1. Read the horizontal scale and vertical scale separately.
  2. Start at the given x-value and project vertically until you meet the graph.
  3. Project horizontally to the y-axis and read the output.
  4. If the point is open, that endpoint is not included.
Worked example

Worked example · Reading the graph

On the first graph, the filled point (−4,3) gives f(−4)=3. The point (2,4) is open, so that piece does not define f(2).

Check the other piece: it starts at x=2 with an open point too, so f(2) is undefined.

Interactive lab

Read a Graph as Evidence

Choose a graph, scan inputs, then check your prediction. Open endpoints are excluded; filled endpoints are included.
-5-4-3-2-101234567-4-3-2-10123456
f(x): 0.00
Which x-values belong to the domain?

Graphs use exact linear interpolation between plotted vertices. Read open circles as excluded and filled circles as included.

Your turn

Which axis is projected onto when finding the domain?
Explanation

02 · Project

Domain, Range & Intercepts

Domain: project the whole graph onto the x-axis. Range: project onto the y-axis. Check every open and closed endpoint before writing interval notation.

Explanation

Mr. Azzam's method · Project onto the axes

  1. For domain, shade every x-value under the graph; for range, shade every output height on the y-axis.
  2. Keep gaps as separate intervals connected by the union symbol.
  3. Use square brackets for included endpoints and parentheses for excluded endpoints.
  4. Write only the x-values for domain and only the y-values for range.
Worked example

Worked example · Disconnected graph

For the first lab graph, the left piece occupies x from −4 to 0 with filled endpoints, and the second occupies x greater than 2 up to 5 inclusive.

Domain: [−4,0] ∪ (2,5]. Range: [−1,3] ∪ 4.

Df={x:f(x) is defined},Rf={f(x):x∈Df}D_f=\{x:f(x)\text{ is defined}\},\qquad R_f=\{f(x):x\in D_f\}

Your turn

A curve starts at an open point with x=-2 and ends at a filled point with x=4, covering all x-values between. What is its domain?
Given question graphIllustrative graph matching the stated information Open circles exclude endpoints; filled circles include points. x horizontal axis; y vertical axis.-3-1135-1012345xy
Illustrative graph matching the stated information
Explanation

03 · Scan

Function Values, Horizontal Lines & Net Change

Explanation

Mr. Azzam's method · Solve from a graph

  1. For f(a), go from x=a to the curve and read the height.
  2. For f(x)=k, draw a horizontal line y=k and record every intersection x-coordinate.
  3. For f(x)>k or f(x)≤k, find the segments above or below the line and transfer them to the x-axis.
  4. Check whether equality is included, and never confuse a point with an interval.
Worked example

Worked example · Four crossings at height zero

The graph below crosses the x-axis at x=−3, −1, 1 and 3. Hence f(x)=0 has four solutions, while f(x)≥0 includes the intervals [−3,−1] and [1,3].

Interactive lab

Horizontal Line · Graphical Inequalities

Move y=k. Count distinct intersection inputs, including a shared vertex only once. Then use the level-zero solution to reason about inequalities.
-4-3-2-101234-3-2-10123k

Try setting k=0. Find every intersection and write the solution of f(x) greater than or equal to zero.

How many distinct inputs solve f(x)=k at this level?
Interactive lab

Zeros and Sign of a Cubic

Investigate f(x) = x³ − 2x² − 5x + 6. Read the zeros and predict where the curve lies on or above the x-axis.
-3-2-101234-15-10-5051015
Where is f(x) ≥ 0?

For f(a)f(a), start at x=a, move vertically to the graph, then read the corresponding y-coordinate. For f(x)=kf(x)=k, intersect the graph with the horizontal line y=k. For inequalities, inspect where the curve is above or below that line.

Net change=f(b)−f(a)\text{Net change}=f(b)-f(a)

Your turn

If f(1)=3f(1)=3 and f(4)=−2f(4)=-2, what is the net change from x=1 to x=4?
Explanation

04 · Compare

Two Functions on One Coordinate Plane

Explanation

Mr. Azzam's method · Compare two outputs

  1. Draw or inspect both functions on the same coordinate system.
  2. Intersections mean f(x)=g(x); read their x-coordinates.
  3. Where the f-graph is higher, f(x)>g(x); where the g-graph is higher, f(x)<g(x).
  4. Split intervals at each intersection and check strict or inclusive endpoints.
Worked example

Worked example · A line and a parabola

For f(x)=2x²+3 and g(x)=5x+6, the graphs intersect at x=−0.5 and x=3.

Between the crossings, the parabola is below the line, so f(x)≤g(x) for −0.5≤x≤3.

Interactive lab

Compare Two Functions

Blue: f(x) = 2x² + 3. Orange: g(x) = 5x + 6. Move the red scanner, compare outputs, then solve the inequality.
-2-101234-50510152025303540

f(x) = 3.00; g(x) = 6.00. g is higher

For which inputs is f(x) ≤ g(x)?

Where the graphs intersect, f(x)=g(x)f(x)=g(x). Where the graph of f lies above g, f(x)>g(x)f(x)>g(x). Always give x-values or x-intervals, not the intersection heights.

Your turn

If two graphs intersect at (2,5)(2,5), which value solves f(x)=g(x)f(x)=g(x) at that point?
Given question graphIllustrative graph matching the stated information Open circles exclude endpoints; filled circles include points. x horizontal axis; y vertical axis.-101234502468(2, 5)xy
fg
Illustrative graph matching the stated information
Explanation

05 · Follow the Curve

Increasing, Decreasing & Constant

Explanation

Mr. Azzam's method · Follow the graph from left to right

  1. Read the x-axis from left to right; ignore whether the graph happens to be above or below zero.
  2. Going up means increasing, going down means decreasing, and a horizontal segment means constant.
  3. Use intervals of x-values, conventionally open intervals for increasing and decreasing.
  4. At a turning point, separate the adjoining intervals.
Interactive lab

Behavior of f(x) = x^(2/3)

Move the scanner and predict the turning behavior. For real inputs, x^(2/3) means the square of the real cube root.
-8-6-4-20246801234
How does the function behave as x passes through zero?

Read from left to right. Rising means increasing, falling means decreasing, and horizontal means constant. State the intervals in terms of x, typically using open intervals for increasing/decreasing behavior.

Your turn

As x increases from 1 to 4, the graph moves downward. The function is...
Given question graphIllustrative graph matching the stated information Open circles exclude endpoints; filled circles include points. x horizontal axis; y vertical axis.0123450123456xy
Illustrative graph matching the stated information
Explanation

06 · Find the Peaks

Maximum and Minimum: Point vs Value

A maximum point gives both coordinates; the maximum value is only the y-coordinate. A local maximum is highest nearby, while an absolute maximum is highest on the entire domain.

Explanation

Mr. Azzam's method · Point versus value

  1. Find the peaks and valleys and read both coordinates carefully.
  2. Report the maximum or minimum point as (x,y); report its value as y only.
  3. Local means highest or lowest nearby; absolute means highest or lowest over the entire domain.
  4. Check endpoints and disconnected pieces before claiming an absolute extreme.
Worked example

Worked example · Quartic peaks and valleys

For f(x)=12x²+4x³−3x⁴, a local maximum point is (−1,5), a local minimum point is (0,0), and the absolute maximum point is (2,32).

The absolute maximum value is 32, not the coordinate pair (2,32).

Interactive lab

Quartic Graph · Turning Points

Investigate f(x) = 12x² + 4x³ − 3x⁴. Move the cursor and distinguish local peaks from the absolute maximum.
-2-10123-40-2002040
Which is the absolute maximum value?
At (a,b):f(a)=b\text{At }(a,b):\quad f(a)=b

Your turn

If a local maximum point is (−1,4)(-1,4), what is the local maximum value?
Given question graphIllustrative graph matching the stated information Open circles exclude endpoints; filled circles include points. x horizontal axis; y vertical axis.-4-3-2-1012-1012345(−1, 4)xy
Illustrative graph matching the stated information
Explanation

07 · Apply

A Graph Tells a Story

Temperature, weight, water levels and traffic capacity: interpret the axes and units, describe increasing/decreasing intervals, locate extremes, and calculate net change in context.

Interactive lab

Real-World Graph Investigation · Reservoir

Read time in hours and depth in metres. Move the scanner, then choose a prediction for each interpretation task. The depth scale starts at 20 m, not zero.
01234567820304050607080Time (hours)Depth (m)

Interpretation task 1 of 6

What is the reservoir depth at hour 4?
Key takeawayMr. Azzam Exam Tip

Always attach units to values, and use intervals of the independent variable when describing trends.

Practice

Guided graph practice

Read, Decide, Then Verify

Each question has its own mathematically scaled graph. Answer before revealing coordinate labels.

Interactive lab

Graph Evidence Practice

Read the plotted coordinates before checking. Coordinate labels are an optional verification tool, not a substitute for your prediction.
-5-4-3-2-1012345-3-2-101234
Read f(-2) from the graph.
Interactive lab

Complete Graph Investigation

One graph, twelve connected questions. Choose a prediction and check it against graphical evidence.
-5-4-3-2-1012345-3-2-10123

Question 1 / 12

Find the domain.
Practice

09 · Mastery

Can You Read Every Detail?

Lesson Mastery Checkpoint

Rate each skill. This is only the starting signal; your actual Practice and Assessment performance will update the plan.

Function values and net change

Domain and range

Intercepts and endpoint notation

Graphical equations and inequalities

Compare two functions

Increasing decreasing and constant intervals

Local and absolute extrema

Interpret real-world graphs

Rate all skills to prepare your personalized practice path.

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