Functions: The Four Core Shapes

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In Grade 10, you learn to sketch and interpret four fundamental graph shapes. Every one is controlled by the same key parameters: $a$ (shape & direction) and $q$ (vertical shift).

The Two Parameters

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The Three Shapes You Must Know

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1. The Straight Line: $y = ax + q$

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• $a > 0$: line goes uphill (left to right). $a < 0$: line goes downhill.
• $q$ = $y$-intercept (where the line crosses the $y$-axis).
• $x$-intercept: let $y = 0$ and solve.

2. The Parabola: $y = ax^2 + q$

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• Turning point is at $(0;\, q)$ — always on the $y$-axis in Grade 10.
• Axis of symmetry is $x = 0$ (the $y$-axis).
• $a > 0$: opens UP (minimum at $q$). $a < 0$: opens DOWN (maximum at $q$).
• $x$-intercepts: let $y = 0$, solve $ax^2 + q = 0$.

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

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• Vertical asymptote at $x = 0$ (the $y$-axis) — no $y$-intercept exists!
• Horizontal asymptote at $y = q$ — the curve approaches but never reaches this value.
• $a > 0$: branches in quadrants I and III. $a < 0$: branches in quadrants II and IV.
• $x$-intercept: let $y = 0$, solve $\frac{a}{x} + q = 0 \Rightarrow x = -\frac{a}{q}$.

4. The Exponential: $y = ab^x + q$

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• Horizontal asymptote at $y = q$ — the curve approaches but never touches this value.
• $b > 1$: growth (graph rises steeply to the right). $0 < b < 1$: decay (graph falls towards asymptote).
• $a > 0$: graph is above the asymptote. $a < 0$: graph is reflected below the asymptote.
• $y$-intercept: let $x = 0$: $y = a + q$ (always exists).
• The graph has no vertical asymptote and no turning point.

Finding Intercepts: The Universal Rules

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🚨 Common Mistakes

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1. Intercept confusion: $y$-intercept → let $x = 0$. $x$-intercept → let $y = 0$. Students constantly mix these up.
2. Hyperbola has NO $y$-intercept: $x = 0$ makes $\frac{a}{x}$ undefined. Don’t try to calculate it.
3. Parabola drawn with a ruler: A parabola is a smooth curve, not a V-shape. Connect points with a flowing curve.
4. Forgetting asymptote labels: Draw asymptotes as dashed lines and write their equations. This is required in exams for both the hyperbola and exponential.
5. Domain/Range errors: The hyperbola’s domain excludes $x = 0$. The parabola’s range depends on the sign of $a$ and the value of $q$. The exponential’s range is restricted by $q$.
6. Exponential $y$-intercept is NOT $q$: The $y$-intercept is $a + q$ (substitute $x = 0$). The asymptote is $y = q$. Don’t confuse them.

💡 Pro Tip: The “Table Mode”

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If you’re struggling to draw a graph, use the “TABLE” mode on your scientific calculator. Enter the formula, pick a start and end $x$ (e.g., $-3$ to $3$), and it gives you a list of coordinates to plot. It’s the ultimate safety net!

🔗 Deep Dives:

• Sketching Graphs (Linear, Quadratic, Hyperbola) — full worked examples for the first three shapes
• The Exponential Graph — growth, decay, asymptotes, and finding equations

📌 Where this leads in Grade 11: Functions — The Logic of Transformation — all four shapes gain a horizontal shift parameter $p$

⏮️ Equations & Inequalities | 🏠 Back to Grade 10 | ⏭️ Finance & Growth