Graph of Functions

Lesson Objectives

Understand the concept of graphing a function on the Cartesian plane.
Identify and sketch graphs of basic functions.
Recognize transformations of functions (translations, reflections, stretches).
Interpret key features of a graph such as intercepts, domain, range, and symmetry.

Lesson Introduction

A graph of a function is a visual representation of the set of all ordered pairs \((x, f(x))\). By analyzing the graph, we can understand the behavior of the function, such as how it increases, decreases, or remains constant.

Lesson Content

Cartesian Plane and Plotting Points

The Cartesian plane consists of two perpendicular axes: the horizontal \(x\)-axis and the vertical \(y\)-axis. A function graph consists of all points \((x, y)\) such that \(y = f(x)\).

Basic Graphs of Functions

Here are the graphs of some common functions:

Linear function: \(f(x) = x\)
Quadratic function: \(f(x) = x^2\)
Cubic function: \(f(x) = x^3\)
Absolute value function: \(f(x) = |x|\)
Square root function: \(f(x) = \sqrt{x}\)

Example 1: Sketch the graph of \(f(x) = x^2\).
Make a table of values:

\(x = -2 \Rightarrow f(x) = 4\)
\(x = -1 \Rightarrow f(x) = 1\)
\(x = 0 \Rightarrow f(x) = 0\)
\(x = 1 \Rightarrow f(x) = 1\)
\(x = 2 \Rightarrow f(x) = 4\)

Plot these points and connect with a smooth curve (a parabola).

Key Features of Graphs

x-intercept(s): Point(s) where the graph crosses the x-axis (\(f(x) = 0\))
y-intercept: Point where the graph crosses the y-axis (\(x = 0\))
Domain: Set of all valid inputs \(x\)
Range: Set of all possible outputs \(f(x)\)
Symmetry: Even (symmetric about y-axis), Odd (symmetric about origin)

Example 2: Identify key features of \(f(x) = x^2\)

x-intercept: \((0, 0)\)
y-intercept: \((0, 0)\)
Domain: \(\mathbb{R}\)
Range: \([0, \infty)\)
Symmetry: Even (symmetric about the y-axis)

Transformations of Functions

Transformations alter the shape or position of the basic graph. Types include:

Vertical shift: \(f(x) + k\) shifts up/down
Horizontal shift: \(f(x - h)\) shifts left/right
Reflection: \(-f(x)\) reflects in x-axis, \(f(-x)\) reflects in y-axis
Stretch/Compression: \(af(x)\) stretches/compresses vertically

Example 3: Describe the transformation from \(f(x) = x^2\) to \(g(x) = (x - 2)^2 + 3\)

Shift right by 2 units (because of \(x - 2\))

Shift up by 3 units (because of \(+3\))

Graphing Piecewise Functions

Piecewise functions have different expressions for different parts of the domain.

Example 4: Graph the function:

\[ f(x) = \begin{cases} x + 2 & \text{if } x < 0 \\[6pt] x^2 & \text{if } 0 \leq x \leq 2 \\[6pt] 4 - x & \text{if } x > 2 \end{cases} \]

To plot this function:

Consider each condition separately.
Plot the segment \(f(x) = x + 2\) for \(x < 0\).
Plot the segment \(f(x) = x^2\) for \(0 \leq x \leq 2\).
Plot the segment \(f(x) = 4 - x\) for \(x > 2\).
Use open or closed circles at the endpoints depending on whether the endpoint is included or excluded.

Exercises

[WAEC] Sketch the graph of \(f(x) = -x^2 + 4\). Identify the vertex and intercepts. [past-question]
Describe the transformation of \(f(x) = |x - 3| + 2\) compared to \(f(x) = |x|\).
State the domain and range of \(f(x) = \sqrt{x - 1}\).
[WASCE] Determine if \(f(x) = x^3\) is odd, even, or neither. [past-question]
Sketch and label the graph of the piecewise function: \[ f(x) = \begin{cases} 2x & \text{if } x \leq 1 \\ x^2 & \text{if } x > 1 \end{cases} \]
Find the x-intercept and y-intercept of the function \(f(x) = 2x - 5\).
Sketch the graph of \(f(x) = \frac{1}{x}\) and describe its asymptotes.
Write the domain and range of \(f(x) = |x + 2|\).
Sketch the graph of \(f(x) = -\sqrt{x + 1}\) and describe the transformation from \(f(x) = \sqrt{x}\).
[NECO] Given \(f(x) = (x + 1)^2 - 4\), find the vertex, axis of symmetry, and intercepts. [past-question]

Conclusion/Recap

A function graph visually shows how input values relate to output values on the coordinate plane. Different types of functions like linear, quadratic, and reciprocal have distinct graph shapes. Graphs reveal key features such as domain, range, intercepts, and symmetry. Transformations can shift, reflect, stretch, or compress a graph. Piecewise functions are graphed in segments, each applying to a specific interval. Even functions are symmetric about the y-axis, while odd functions are symmetric about the origin. Graphing functions is essential for analyzing behavior and solving real-world problems.