IB Mathematics: Applications and Interpretation HL 6 Sections80 LessonsLifetime
Graphs and Key features of functions AI HL
Lesson 2.3 · Graphs and Key Features of Functions
Reading, sketching and using graphs to understand relationships.
Faith connection
Ruth showed practical love by staying with Naomi when Naomi faced an uncertain future. Ruth said, “whither thou goest, I will go” (Ruth 1:16, KJV). As we study a graph, we look beyond one isolated point to understand a whole relationship. In our relationships, love likewise calls us to notice someone’s circumstances and remain faithful through change. Jesus is our Savior and Lord; we can come to him for the love and strength to care for others.
Learning objectives
By the end of this lesson, you should be able to:
- interpret the graph of
and distinguish a careful draw from a sketch; - sketch a graph from its equation, key features or a real-world context;
- identify intercepts, zeros, maxima, minima, vertices, symmetry and asymptotes;
- use graphing technology to investigate functions, sums, differences and intersections;
- interpret key features and intersections in context.
1. What does a graph show?
The graph of consists of points 
for inputs 
in the domain. An -intercept occurs where 
; a 
-intercept occurs at 
, provided 
belongs to the domain. The graph also shows where outputs increase, decrease or stay constant.
Draw versus sketch: To draw a graph, plot it accurately using a suitable scale. To sketch, show its correct overall shape and label the axes and relevant key features; exact plotting of many points is unnecessary.
.
![\[f(x)=(x-1)(x-3)=(x-2)^2-1\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-e71481f14f8fe5d135edeccdb558b72b_l3.png?resize=317%2C23&ssl=1 "Rendered by QuickLaTeX.com")
The -intercepts are 
and 
, the -intercept is 
, and the minimum vertex is 
. Draw a smooth upward-opening parabola through these points, with axis of symmetry 
. Label both axes and all four features.
2. Finding and describing key features
| Feature | Meaning or method |
|---|---|
| Zero / root | An input for which . The corresponding point is an -intercept. |
| Maximum / minimum | The greatest / least output over the stated domain, if it exists. Specify whether it is local or absolute when relevant. |
| Vertex | The turning point of a parabola; its axis of symmetry passes through it. |
| Symmetry | Even:  , symmetry about the -axis. Odd:  , symmetry about the origin. |
| Vertical asymptote | A line  approached as outputs grow without bound near  . |
| Horizontal asymptote | A line  approached as grows toward positive or negative infinity. |
.
The denominator cannot be zero, so the domain excludes 
and the vertical asymptote is . As becomes very large in magnitude, 
approaches , giving horizontal asymptote 
.
![\[g(0)=1,\qquad 0=\frac{2}{x-1}+3\ \Longrightarrow\ x=\frac13\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-bb7f801c0d26908a9003bc628c932a25_l3.png?resize=351%2C41&ssl=1 "Rendered by QuickLaTeX.com")
Thus the intercepts are 
and 
. The graph is a shifted reciprocal curve with one branch above for 
and one below for 
. Its range excludes 
.
Watch the domain: A window on a calculator does not determine the domain. A graph can also have an apparent maximum on the screen without having an absolute maximum.
3. Sketching from a context
Read the units and allowed inputs before plotting. Decide whether the graph is continuous, identify important points, and choose scales that show the behaviour clearly. A physically meaningful domain can restrict a mathematical graph.
after 
seconds. Sketch its height from release until it hits the ground.
![\[h(t)=-5(t-2)^2+21\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-7e4b8a61c85f6d51d6708e8b44596a04_l3.png?resize=195%2C23&ssl=1 "Rendered by QuickLaTeX.com")
The ball starts at and reaches its maximum height of 
m at 
s. Setting 
gives 
s; the other root is negative and does not belong to the context. Sketch only 
, label both axes with units, and mark , 
and approximately 
.
4. Graphs of sums and differences
For functions 
and 
, the sum and difference combine their outputs at the same input:
![\[(f+g)(x)=f(x)+g(x),\qquad(f-g)(x)=f(x)-g(x).\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-0e1ead05b408562c1cc52f1fa3dcd5eb_l3.png?resize=485%2C20&ssl=1 "Rendered by QuickLaTeX.com")
The common domain contains only inputs allowed by both functions. At an intersection of and , their difference is zero.
and 
.
![\[(f-g)(x)=x^2-2x-3=(x-3)(x+1).\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-5b00bb7a85adf320ce3e01deb51e88df_l3.png?resize=369%2C23&ssl=1 "Rendered by QuickLaTeX.com")
The zeros of 
are 
and 
, so and intersect at 
and 
. Moreover, 
.
5. Using graphing technology
- Enter the functions as
and 
; enter 
or 
when needed. - Set a viewing window that includes relevant inputs and outputs. Inspect the graph for intercepts, turning points and possible asymptotes.
- Use the calculator’s zero, minimum/maximum or intersection command to obtain coordinates. Check that the selected feature is the intended one.
- Verify numerical results in the original equations, label the graph, and round only the final answer as requested.
Exploring parameters: Sliders can show how coefficients change a graph. For example, in 
, varying 
moves the vertex horizontally, varying 
moves it vertically, and changing the sign of reverses its opening direction.
and 
at . Check it: 
and 
. Both graphs have the same output, so the point is an intersection.
IB-style Paper 1 practice
Show your reasoning. Use graphing technology where it is requested, and give coordinates to three significant figures when necessary.
1. Let 
.
- Find both -intercepts and the -intercept.
- Find the vertex and state the range of for
. - Describe a labelled sketch.
Show solution
. The -intercepts are and 
; the -intercept is 
. The vertex is 
, so the range is 
. Sketch an upward-opening parabola with axis , labelling these points and both axes.
2. For 
, state the domain, range, asymptotes and intercepts.
Show solution
Domain: 
; range: 
. Vertical asymptote: 
; horizontal asymptote: 
. Since 
, the -intercept is . Solving 
gives the -intercept 
.
3. Let 
and 
.
- Find
and 
. - Find every intersection of the graphs of and .
Show solution
and 
. Hence the intersection inputs are and . Substitution into gives 
and 
.
4. The function 
is restricted to 
. Find its absolute maximum and minimum values on this domain. Explain why checking the vertex alone is insufficient.
Show solution
, so the maximum is 
at . At the endpoints, 
and 
; therefore the minimum is at 
. An absolute extremum on a restricted closed interval can occur at an endpoint.
5. A function is defined by 
. Determine whether it is even, odd or neither. Find its zeros and state the symmetry to show on a sketch.
Show solution
, so 
is odd and its graph has origin symmetry. Solving 
gives zeros 
, and 
.
6. Using graphing technology, estimate all intersection points of 
and 
to three significant figures. Explain which calculator graph can be used to check the number of intersections.
Show solution
Intersections solve 
. A calculator gives approximately 
. Substituting into gives 
, 
and 
, to three significant figures. Graphing 
shows three zeros corresponding to the three intersections.
Applications and interpretation
A. Break-even points. A small business models revenue and cost, in hundreds of euros, by 
and 
, where is the number of hundreds of items sold and 
.
- Write a function for profit
. - Find both break-even sales quantities.
- Find the greatest predicted profit and interpret its units.
Show solution
. Setting 
gives 
, hence 
or 
. These represent about 
or 
items according to the continuous model; actual whole-item quantities need a practical rounding decision. The vertex occurs at 
(1000 items), where 
hundreds of euros, or €3400.
B. Meeting on a route. Two cyclists’ distances from a starting point, in kilometres after hours, are 
and 
, for 
. Find when they are at the same distance and interpret both intersections.
Show solution
Set 
, giving 
. At 
both are at the start; at 
hours both are 
km from it. The graph intersections are 
and 
.
C. Medicine model. A model for the concentration of a medicine is 
milligrams per litre for 
hours. Use graphing technology to estimate the highest concentration and the time at which it occurs. Describe the graph’s long-term behaviour.
Show solution
The maximum occurs at hours and equals 
mg/L. The graph begins at , rises to 
and then falls toward the horizontal asymptote 
as increases. The model predicts concentrations approaching zero over time.
D. Interpreting a restricted graph. A park models the number of visitors, in hundreds, hours after opening by 
for 
. Find the opening value, the peak value and time, and the closing value. State the range over the stated interval.
Show solution
, so the model predicts 200 visitors at opening. Its maximum is 
, or 1800 visitors, four hours after opening. 
, or 200 visitors, at closing. The range on is 
, measured in hundreds of visitors.
Paper 1 checklist
Label axes and units; mark intercepts and turning points; show asymptotes with their equations; state the relevant domain; distinguish a root’s -value from its intercept point; and interpret calculator answers in the context of the question.