IB Mathematics AA SL The graph of linear equation function Study Notes

IB Mathematics AA SL The graph of linear equation function Study Notes

IB Mathematics AA SL The graph of linear equation function Study Notes Offer a clear explanation of The graph of linear equation function , including various formula, rules, exam style questions as example to explain the topics. Worked Out  examples and common problem types provided here will be sufficient to cover for topic The graph of linear equation function

Graph of a Function and Command Terms

The Equation \( y = f(x) \)

A function is a rule that assigns to each input \( x \) exactly one output \( y \). This relationship is often written as: \( y = f(x) \), where:

  • \( x \) is the input or independent variable (usually on the horizontal axis).
  • \( y \) is the output or dependent variable (plotted on the vertical axis).
  • \( f \) is the name of the function or rule applied to \( x \).

For example, if \( f(x) = x^2 \), then the graph of \( y = f(x) = x^2 \) is a parabola.

Graph of a Function

The graph of a function is a visual representation of all points \( (x, f(x)) \). It helps us:

  • See how the output changes as the input varies
  • Identify key features: intercepts, turning points, symmetry, etc.
  • Understand behavior like increasing, decreasing, and discontinuities

Difference Between “Draw” and “Sketch”

TermMeaningWhat to Include
DrawAccurate diagram to scale using tools or technologyCorrect scale, axis labels, points plotted precisely, often requires ruler or graphing software.
SketchFreehand graph that captures the essential shape or behaviorKey features (intercepts, turning points, asymptotes, etc.), but no need for perfect scale or precision.

Note: When asked to “draw,” use a ruler or calculator. When asked to “sketch,” focus on the general behavior of the function.

Example :

Draw the graph of the function \( f(x) = x^2 – 4x + 3 \)

▶️ Answer/Explanation
  • Plot at least 5 points using a table of values.
  • Find the vertex by completing the square or using \( x = \frac{-b}{2a} \).
  • Mark the x-intercepts: \( x = 1 \) and \( x = 3 \).
  • Mark the y-intercept: \( f(0) = 3 \).
  • Draw on a clearly labeled Cartesian grid with appropriate scale.

Example : 

Sketch the graph of the function \( f(x) = x^2 – 4x + 3 \)

▶️ Answer/Explanation
  • Indicate the general U-shape of the parabola.
  • Label the vertex roughly at \( x = 2 \), \( y = -1 \).
  • Mark approximate intercepts on x-axis and y-axis.
  • Clearly label the axes but exact scale is not necessary.
  • No need to calculate precise values.

Creating a Sketch from Information or Context

Creating a Sketch from Information or Context

A sketch is a freehand drawing of a graph that illustrates the key features and general shape of a function. You may be asked to sketch a graph based on algebraic information, a word problem, or from a graph displayed on a screen (e.g., calculator or software).

Purpose of a Sketch

A sketch helps to visualize:

  • The shape and key characteristics of a function
  • The behavior of the function at certain points
  • The general trend without requiring perfect scale or plotting

 Key Features to Include in a Sketch

Even though a sketch is not drawn to scale, it should include:

  • Axes: Clearly label x-axis and y-axis with appropriate scale/units if needed
  • Intercepts: Points where the graph crosses the x-axis and y-axis
  • Turning Points / Maxima / Minima: Indicate any known stationary points
  • Asymptotes: Vertical, horizontal, or oblique asymptotes where applicable
  • End Behavior: Indicate how the graph behaves as \( x \to \infty \) or \( x \to -\infty \)
  • Domain and Range: If restricted, reflect in the sketch (e.g., open/closed circles)

 Transferring a Graph from a Screen (e.g., GDC or Software)

When copying a graph from a Graphing Display Calculator (GDC) or software like Desmos or GeoGebra to paper:

  1. Note down the x- and y-intercepts
  2. Identify turning points (use the GDC “maximum” or “minimum” tools)
  3. Identify asymptotes (using trace or table function)
  4. Observe how the graph behaves as \( x \to \infty \) and \( x \to -\infty \)
  5. Draw the axes and plot the intercepts and key points
  6. Sketch a smooth curve connecting the points in the correct shape

Example : 

Sketch the graph of \( f(x) = x^2 – 4x + 3 \).

▶️ Answer/Explanation
  • Intercepts: Factor to get \( (x-1)(x-3) \). x-intercepts at \( x = 1 \), \( x = 3 \). y-intercept: \( f(0) = 3 \).
  • Vertex: \( x = \frac{1+3}{2} = 2 \), \( f(2) = -1 \). Vertex at (2, -1).
  • Sketch shape: Parabola opening upward with axis of symmetry \( x = 2 \).

Example :

Sketch the graph of \( f(x) = \frac{1}{x-3} \).

▶️ Answer/Explanation
  • Vertical Asymptote: \( x = 3 \)
  • Horizontal Asymptote: \( y = 0 \)
  • Key point: \( f(2) = -1 \), \( f(4) = 1 \)
  • End behavior: Approaches 0 as \( x \to \pm \infty \)

Example : 

Sketch the graph of \( f(x) = |x – 2| \).

▶️ Answer/Explanation
  • Vertex: At \( x = 2 \), \( f(2) = 0 \)
  • Shape: V-shape opening upward
  • Additional points: \( f(1) = 1 \), \( f(3) = 1 \)

 

Using Technology to Graph Functions, Sums, and Differences

Using Technology to Graph Functions, Sums, and Differences

Graphing technology such as a GDC (Graphical Display Calculator), Desmos, or GeoGebra can be used to visualize functions and their combinations.

Common Tasks with Technology:

  • Plot individual functions like \( f(x) = x^2 \), \( g(x) = 2x \)
  • Graph sums: \( f(x) + g(x) \)
  • Graph differences: \( f(x) – g(x) \)
  • Compare how each function behaves and how they combine.

Steps Using Desmos (or GDC):

  1. Open Desmos (or your GDC graph function screen)
  2. Enter the functions one by one:
    • f(x) = x^2
    • g(x) = 2x
    • f(x) + g(x) → Desmos automatically shows the result
    • f(x) - g(x)
  3. Observe how the graphs of \( f(x) + g(x) \) and \( f(x) – g(x) \) relate to \( f(x) \) and \( g(x) \)
  4. You can also compare using different color graphs

Interpretation:

  • The graph of \( f(x) + g(x) \) represents the pointwise sum of the y-values of \( f(x) \) and \( g(x) \) at each x. Similarly, \( f(x) – g(x) \) subtracts the y-values of \( g(x) \) from \( f(x) \) at each x.
  • If \( f(x) \) and \( g(x) \) are increasing, then \( f(x) + g(x) \) typically increases faster.
  • If \( f(x) = g(x) \), then \( f(x) – g(x) = 0 \) for all x.
  • The shape of the resulting graph is influenced by the dominant function.

Example : Graph the sum of the functions from

  • f(x) = x²
  • g(x) = 6x + 9

Function: h(x) = f(x) + g(x) 

▶️ Answer/Explanation

h(x) = f(x) + g(x) = x² + 6x + 9

Use your GDC to enter:

  • $Y = x^2 + 6x + 9$

This is a new parabola (perfect square trinomial):

Vertex: (-3, 0)   Opens: Upward

 

Example : Graph the difference of the functions from 

  • f(x) = $x^2$
  • g(x) = $2x + 1$

Function: k(x) = f(x) – g(x) 

▶️ Answer/Explanation

k(x) = f(x) – g(x) = $x^2- 2x – 1$

Use your GDC to enter:

  • $Y = x^2- 2x – 1$

This is a parabola:

Vertex: (1, -2)   Opens: Upward

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