IB Mathematics AI SL Determine key features of graphs MAI Study Notes - New Syllabus

IB Mathematics AI SL Determine key features of graphs MAI Study Notes

LEARNING OBJECTIVE

Determine key features of graphs

Key Concepts:

Properties of Graphs

Maximum and Minimum Values

Definition:

Maximum: The highest point (y-value) of a function within a given domain.
Minimum: The lowest point (y-value) of a function within a given domain.

Types:

Global (Absolute): The highest/lowest point over the entire domain.
Local (Relative): The highest/lowest point within a small interval (but not necessarily the entire graph).

Example:

For $ f(x) = -x^2 + 4x + 5 $, the vertex (highest point) is at (2, 9).

Intercepts

X-intercepts (Roots):

Where $ y = 0 $ (graph crosses the x-axis).
Found by solving $ f(x) = 0 $.

Y-intercepts:

Where $ x = 0 $ (graph crosses the y-axis).
Found by evaluating $ f(0) $.

Example

For $ f(x) = x^2 – 4x + 3 $: Find x and y intercept.

▶️Answer/Explanation

Solution

x-intercepts: $ (1, 0) $ and $ (3, 0) $ (solutions to $ x^2 – 4x + 3 = 0 $).

y-intercept: $ (0, 3) $ (since $ f(0) = 3 $).

Symmetry

Even Functions:

Symmetric about the y-axis ($ f(-x) = f(x) $).

Example:

$ f(x) = x^2 $.

Odd Functions:

Symmetric about the origin ($ f(-x) = -f(x) $).

Example:

$ f(x) = x^3 $.

Asymptotes (Vertical & Horizontal)

Vertical Asymptotes:

Occur where a function is undefined (e.g., denominator = 0).

Example:

$ f(x) = \frac{1}{x-3} $ has a vertical asymptote at $ x = 3 $.

Horizontal Asymptotes:

Describe end behavior as $ x \rightarrow \pm \infty $.

Example:

$ f(x) = \frac{2x}{x+1} $ approaches $ y = 2 $.

Using Graphing Technology

Key Features to Check:

Zeros (Roots): Where $ f(x) = 0 $.
Extrema: Max/min points.
Intersections: Points where two graphs meet.

Effects of Parameters (Transformations)
General Form:

$ f(x) = a(x – h)^2 + k $ (for quadratics).

Impact of Parameters:

a: Controls width and direction (up/down).
h: Shifts graph left/right.
k: Shifts graph up/down.

Example

Consider the functions

$ f(x) = (x – 3)^2 – 4 $

Define all the Key features of functions.

▶️Answer/Explanation

Solution:

Graphing $f(x) = (x – 3)^2 – 4$ on any graphing calculator (e.g., Desmos, GDC):

Vertex at $(3, -4)$,
Parabola opens upward,
X-intercepts at 1 and 5,
Y-intercept at 5,
Symmetry about $x = 3$

Quadratic functions do not have asymptotes.

$ \begin{aligned}
f(x) = g(x) &\Leftrightarrow (x – 3)^2 – 4 = x – 5 \Leftrightarrow x^2 – 6x + 9 – 4 = x – 5 \Leftrightarrow x^2 – 7x + 10 = 0 \\
&\Leftrightarrow x = 2 \text{ or } x = 5
\end{aligned} $

By using either $ f(x) $ or $ g(x) $ we find $ y = -3 $, $ y = 0 $ respectively.

Hence, the curves intersect at points $ (2, -3) $ and $ (5, 0) $

SOLVING EQUATIONS AND INEQUALITIES BY USING GRAPHS

We can solve equations of the form $ f(x) = g(x) $ inequalities of the form $ f(x) > g(x) $ or $ f(x) \geq g(x) $ by using GDC – graph mode

METHOD A: we find the intersection points of the graphs

$ \begin{aligned}
y_1 &= f(x) \\
y_2 &= g(x)
\end{aligned} $

Solutions of $ f(x) = g(x) $: $ x $-coordinates of intersection points
Solutions of $ f(x) > g(x) $: intervals where $ y_1 = f(x) $ is above $ y_2 = g(x) $

METHOD B: we find the roots of the graph

$ y_1 = f(x) – g(x) $

Solutions of $ f(x) – g(x) = 0 $: the roots of the graph
Solutions of $ f(x) – g(x) > 0 $: intervals where $ y_1 = f(x) – g(x) $ is positive

Example

Consider

$ f(x) = (x – 3)^2 – 4 \text{ and } g(x) = x – 5 \text{. } $

a) Solve the equation $ f(x) = g(x) $.

b) Solve the inequality $ f(x) > g(x) $.

▶️Answer/Explanation

Solution:

(a)METHOD A: Look at the graphs of $ y_1 = f(x) $ and $ y_2 = g(x) $

. The intersection points occur at $ x = 2 $, $ x = 5 $