IB Mathematics AA SL Key features of graphs Study Notes

Key features of graph

Introduction to Functions and Graphs

A function is a mathematical relationship where each input \( x \) corresponds to exactly one output \( y \), expressed as \( y = f(x) \). The graph of a function represents this relationship visually, showcasing how \( y \) changes with \( x \).

Key Aspects of Graphing a Function

• Domain: The set of all possible input values (\( x \)) that the function can accept. For example, the domain of \( f(x) = \sqrt{x} \) is \( x \ge 0 \).
• Range: The set of all possible output values (\( y \)). For \( f(x) = x^2 \), the range is \( y \ge 0 \).
• Intercepts: Points where the graph crosses the \( x \)-axis (roots) and the \( y \)-axis (when \( x = 0 \)).
• Key Features: Include maximum and minimum points, turning points, and asymptotes.

Roots of a Function

The roots (or zeros) of a function are the \( x \)-values where the function equals zero (\( f(x) = 0 \)). These are the points where the graph intersects the \( x \)-axis. For example, the roots of \( f(x) = x^2 – 4 \) are \( x = \pm 2 \).

• Finding Roots: Solving \( f(x) = 0 \) can yield the roots. For quadratic functions like \( ax^2 + bx + c = 0 \), use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \).
• Graphical Interpretation: On the graph of a function, the roots are the \( x \)-coordinates where the graph touches or crosses the \( x \)-axis.
• Multiplicity: A root’s multiplicity indicates how many times it repeats. If \( (x – a)^2 \) is a factor, \( x = a \) is a root of multiplicity 2, causing the graph to touch but not cross the \( x \)-axis.

Difference Between “Draw” and “Sketch”

• Draw: This involves creating an accurate and precise representation of the graph on graph paper, with correctly scaled axes and plotted points. It requires exact calculations and labeling of key features like intercepts and turning points.
• Sketch: A rough representation of the graph that shows the general shape and important features without precise scaling. It highlights the overall trend, intercepts, and key points but does not require exact plotting.

Example: For the function \( y = x^2 – 4 \):

• When asked to draw the graph, you would precisely plot points like (-2, 0), (0, -4), and (2, 0).

• When asked to sketch, you would illustrate the U-shape of the parabola with intercepts at \( x = \pm 2 \) and a minimum at \( (0, -4) \).

Graphing Functions Using Technology

Graphing calculators, software like Desmos or GeoGebra, and online tools can help visualize functions quickly. These tools allow you to:

• Plot standard and complex functions (e.g., \( y = \sin(x) \), \( y = \ln(x) \), or polynomial functions).
• Identify key features like intercepts, maximum and minimum points, and asymptotes.
• Graph transformations, such as shifts, reflections, and stretches.

Creating Sketches from Given Information

• Identify intercepts, maximums, and minimums.
• Analyze the function’s behavior as \( x \to \infty \) and \( x \to -\infty \).
• Check for symmetry or any specific characteristics like periodicity for trigonometric functions.

Example: Sketch the graph of \( y = |x| – 2 \). It forms a V-shape with a vertex at \( (0, -2) \) and opens upwards.

Graphing Sums and Differences of Functions

• Sum: \( h(x) = f(x) + g(x) \)
• Difference: \( h(x) = f(x) – g(x) \)

Example: For \( f(x) = x^2 \) and \( g(x) = 2x \):

• The sum \( h(x) = x^2 + 2x \) represents a parabola opening upwards.

• The difference \( h(x) = x^2 – 2x \) is a shifted parabola.

Labelling Axes and Identifying Key Features

• Axes: Label both \( x \)- and \( y \)-axes with appropriate scales.
• Intercepts: Mark where the graph crosses the axes.
• Turning Points: Indicate points where the graph changes direction.
• Asymptotes: Draw and label any vertical or horizontal asymptotes.

Example: For \( y = \frac{1}{x} \):

• Label the vertical asymptote at \( x = 0 \) and the horizontal asymptote at \( y = 0 \).