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Graph of a Function

CSCA Graph of a Function study guide organized around the publicly available CSCA syllabus. Practice Mathematics questions on aicsca.com.

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Graph of a Function

1. Concept

The **Graph** of a function is the visual representation of the relationship $y=f(x)$ on a coordinate plane.

Visualizing the graph helps identify key properties:

* **Domain/Range**: The extent along the x and y axes.

* **Monotonicity**: Where the graph rises or falls.

* **Symmetry**: Even functions (y-axis symmetry) or Odd functions (origin symmetry).

* **Intercepts**: Points where the graph crosses the axes.

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2. Core Graphing Methods

#### (1) Point-Plotting Method

Used for unknown functions. Steps: **Table** (select key points), **Plot**, **Connect**.

* **Key Points**: Zeros, extrema, endpoints.

#### (2) Transformation Method (High Efficiency)

Start with a basic function (e.g., $y=x^2, \sin x, e^x$) and apply geometric transformations.

**Rules of Thumb**:

* **Translation (Shift)**:

* **Horizontal**: $y = f(x+h)$ shifts **Left** by $h$; $y = f(x-h)$ shifts **Right** by $h$ ($h>0$). (Remember: "Left Add, Right Subtract")

* **Vertical**: $y = f(x) + k$ shifts **Up** by $k$; $y = f(x) - k$ shifts **Down** by $k$ ($k>0$).

* **Scaling (Stretch/Compress)**:

* $y = Af(x)$: Vertical stretch ($A>1$) or compression ($0<A<1$).

* $y = f(\omega x)$: Horizontal compression ($\omega>1$) or stretch ($0<\omega<1$).

* **Reflection**:

* $y = -f(x)$: Reflect over x-axis.

* $y = f(-x)$: Reflect over y-axis.

* $y = |f(x)|$: Reflect parts below x-axis to above.

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3. Typical Examples

**Example 1: Trigonometric Transformation**

Sketch $y = 2\sin(3x - \frac{\pi}{2}) + 1$.

**Solution**:

**Step 1: Standardize**. Factor out the coefficient: $y = 2\sin[3(x - \frac{\pi}{6})] + 1$.

**Step 2: Sequence**.

1. **Base**: $y = \sin x$

2. **Horizontal Compress**: $y = \sin(3x)$ (Period becomes $2\pi/3$)

3. **Horizontal Shift**: Shift **Right** by $\frac{\pi}{6}$ $\rightarrow$ $y = \sin[3(x - \frac{\pi}{6})]$

4. **Vertical Stretch**: Amplitude becomes 2 $\rightarrow$ $y = 2\sin[3(x - \frac{\pi}{6})]$

5. **Vertical Shift**: Up by 1.

**Example 2: Rational Function with Asymptotes**

Sketch $f(x) = \frac{x+1}{x-2}$.

**Solution**:

1. **Simplify**: $f(x) = 1 + \frac{3}{x-2}$.

2. **Transform**: Shift $y = \frac{3}{x}$ Right by 2, Up by 1.

3. **Asymptotes**:

* Vertical: $x = 2$ (Denominator is 0).

* Horizontal: $y = 1$ (Limit as $x \to \infty$).

4. **Intercepts**: $(0, -0.5)$ and $(-1, 0)$.

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4. Common Pitfalls

* **Order of Operations**: Especially in trig functions. Always factor out $\omega$ first (i.e., $f(\omega(x+\phi))$) to see the true horizontal shift.

* **Missing Asymptotes**: Always draw asymptotes as dashed lines first for rational, exponential, or logarithmic functions.