Sketching Graphs: Linear, Quadratic & Hyperbola

The Logic of Functions

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A function is a rule that takes an input ($x$) and produces exactly one output ($y$). Every function has a shape, and that shape is controlled by the formula.

In Grade 10, you need to master three shapes:

1. The Linear Function: $y = ax + q$

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What the parameters control

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• $a$ (gradient): How steep the line is, and which direction it goes.
  • $a > 0$: Line goes UP from left to right (increasing)
  • $a < 0$: Line goes DOWN from left to right (decreasing)
  • $a = 0$: Horizontal line
• $q$ (y-intercept): Where the line crosses the $y$-axis.

How to Sketch

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1. Plot the y-intercept: The point $(0; q)$.
2. Use the gradient to find a second point: From the y-intercept, go “rise over run”. If $a = \frac{2}{3}$, go up 2 and right 3.
3. Draw the line through both points with a ruler.

Worked Example

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Sketch $y = -2x + 3$

• $q = 3$: y-intercept at $(0; 3)$
• $a = -2 = \frac{-2}{1}$: From $(0; 3)$, go down 2, right 1 → $(1; 1)$
• x-intercept: Let $y = 0$: $0 = -2x + 3 \Rightarrow x = \frac{3}{2}$

Finding the Equation

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If given gradient $a = 3$ and point $(2; 8)$:

$y = ax + q$

$8 = 3(2) + q$

$q = 2$

$y = 3x + 2$

2. The Quadratic Function (Parabola): $y = ax^2 + q$

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What the parameters control

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• $a$ (shape & direction):
  • $a > 0$: Opens UPWARD (“happy face”) — minimum turning point
  • $a < 0$: Opens DOWNWARD (“sad face”) — maximum turning point
  • Large $|a|$: Narrow parabola (steep sides)
  • Small $|a|$: Wide parabola (gentle sides)
• $q$ (vertical shift): Moves the parabola up or down. The turning point is at $(0; q)$.

How to Sketch

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1. Turning point: Always at $(0; q)$ for this form.
2. y-intercept: Same as the turning point: $(0; q)$.
3. x-intercepts: Let $y = 0$ and solve $ax^2 + q = 0$.
4. Shape: Check the sign of $a$.
5. Extra points: Use a table of values ($x = -2, -1, 0, 1, 2$).

Worked Example

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Sketch $y = 2x^2 - 8$

Step 1: Turning point at $(0; -8)$

Step 2: $a = 2 > 0$, so it opens upward.

Step 3: x-intercepts: $0 = 2x^2 - 8 \Rightarrow x^2 = 4 \Rightarrow x = \pm 2$

Points: $(-2; 0)$ and $(2; 0)$

Step 4: Table of values:

Plot these points and draw a smooth curve (NOT with a ruler!).

Key Properties

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• Domain: $x \in \mathbb{R}$ (all real numbers) — always
• Range: If $a > 0$: $y \geq q$. If $a < 0$: $y \leq q$.
• Axis of symmetry: $x = 0$ (the y-axis)
• Increasing/Decreasing: If $a > 0$: decreasing for $x < 0$, increasing for $x > 0$.

Finding the Equation

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If given the turning point $(0; -3)$ and point $(2; 5)$:

$y = ax^2 + q$

$5 = a(4) + (-3)$

$a = 2$

$y = 2x^2 - 3$

3. The Hyperbola: $y = \frac{a}{x} + q$

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What the parameters control

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• $a$ (shape & quadrants):
  • $a > 0$: Branches in Quadrants 1 and 3 (top-right and bottom-left)
  • $a < 0$: Branches in Quadrants 2 and 4 (top-left and bottom-right)
• $q$ (horizontal asymptote): The horizontal line the graph approaches but never touches.

The Two Asymptotes

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Every hyperbola has TWO invisible boundary lines:

• Vertical asymptote: $x = 0$ (the y-axis) — because $\frac{a}{0}$ is undefined
• Horizontal asymptote: $y = q$

How to Sketch

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1. Draw the asymptotes as dotted lines first.
2. Find the y-intercept: There is NONE (because $x = 0$ is the vertical asymptote).
3. Find the x-intercept: Let $y = 0$: $0 = \frac{a}{x} + q \Rightarrow x = -\frac{a}{q}$.
4. Plot extra points using a table of values.
5. Draw smooth curves that approach but never touch the asymptotes.

Worked Example

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Sketch $y = \frac{4}{x} - 1$

Step 1: Asymptotes: $x = 0$ (vertical), $y = -1$ (horizontal)

Step 2: $a = 4 > 0$, so branches in Q1 and Q3 (relative to asymptotes).

Step 3: x-intercept: $0 = \frac{4}{x} - 1 \Rightarrow \frac{4}{x} = 1 \Rightarrow x = 4$

Step 4: Table:

Key Properties

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• Domain: $x \in \mathbb{R}$, $x \neq 0$
• Range: $y \in \mathbb{R}$, $y \neq q$
• No turning point — the graph is always increasing or always decreasing in each branch
• Lines of symmetry: $y = x + q$ and $y = -x + q$

4. Reading Information from Graphs

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Exam questions often give you a graph and ask you to determine:

🚨 Common Mistakes

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1. Intercept confusion: y-intercept means $x = 0$. x-intercept means $y = 0$. Students swap these constantly.
2. Parabola with a ruler: A parabola is a CURVE. Never connect points with straight lines.
3. Hyperbola crossing asymptotes: The graph NEVER touches or crosses an asymptote. If your sketch does, something is wrong.
4. Forgetting asymptote labels: In exams, you MUST draw asymptotes as dotted lines AND label them (e.g., $y = -1$).
5. Domain/Range confusion: Domain is the set of $x$-values (horizontal). Range is the set of $y$-values (vertical).

💡 Pro Tip: The Table Method

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If you’re unsure about a graph’s shape, make a table with $x = -3, -2, -1, 0, 1, 2, 3$ and calculate each $y$-value. Plot the points and connect them. This works for ANY function and is your ultimate safety net.

🔗 Related Grade 10 topics:

• Solving Equations — finding x-intercepts means solving $f(x) = 0$
• Factorization — factoring $ax^2 + q = 0$ gives x-intercepts of the parabola
• Linear Patterns — the general term $T_n = dn + c$ is a linear function
• Simple & Compound Interest — compound growth is an exponential function in action

📌 Where this leads in Grade 11 — the same three functions gain horizontal shifts:

• The Parabola — turning point form $y = a(x-p)^2 + q$, completing the square
• The Hyperbola — shifted asymptotes $y = \frac{a}{x-p} + q$
• The Exponential — growth, decay, and the horizontal asymptote

🏠 Back to Functions