Sketching curves

1. Overview

Sketching curves is a core skill for IGCSE Mathematics. It means drawing the approximate shape of a function's graph, indicating key features like intercepts, turning points, and asymptotes. You won't plot every point, but you will show the essential behavior of the function. This skill helps visualize algebraic relationships and solve problems graphically.

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Key Definitions

• Intercept: The point where a graph crosses an axis. The $y$-intercept occurs where $x = 0$, and $x$-intercepts (roots) occur where $y = 0$.
• Gradient ($m$): The steepness of a line, calculated as $\frac{\text{change in } y}{\text{change in } x}$.
• Parabola: The characteristic U-shaped or n-shaped curve of a quadratic function.
• Turning Point (Vertex): The maximum or minimum point of a quadratic or cubic curve.
• Asymptote: A straight line that a curve approaches increasingly closely but never actually touches.
• Sketch: A freehand drawing showing the correct general shape and labeling all key coordinates ($x$-intercepts, $y$-intercepts, and turning points).

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Core Content

(a) Linear Graphs ($y = mx + c$)

Linear graphs are always straight lines.

• $m$ is the gradient.
• $c$ is the $y$-intercept.

Method to Sketch:

1. Identify the $y$-intercept $(0, c)$.
2. Find the $x$-intercept by setting $y = 0$ and solving for $x$.
3. Draw a straight line through both points using a ruler.

Worked example 1 — Sketching a linear graph

Question: Sketch the graph of the line $y = 3x + 6$.

1. Find the $y$-intercept: Set $x = 0$. $y = 3(0) + 6$ $y = 6$ The $y$-intercept is at the point $(0, 6)$.

2. Find the $x$-intercept: Set $y = 0$. $0 = 3x + 6$ Subtract 6 from both sides: $-6 = 3x$ Divide both sides by 3: $x = -2$ The $x$-intercept is at the point $(-2, 0)$.

3. Sketch: Draw a straight line through the points $(0, 6)$ and $(-2, 0)$.

(b) Quadratic Graphs ($y = ax^2 + bx + c$)

Quadratic graphs are parabolas.

• If $a > 0$, the curve is a U-shape (minimum).
• If $a < 0$, the curve is an n-shape (maximum).

Method to Sketch:

1. Shape: Look at the sign of $x^2$.
2. $y$-intercept: Set $x = 0$. The intercept is $(0, c)$.
3. $x$-intercepts (Roots): Set $y = 0$ and factorize or use the quadratic formula.
4. Turning Point: The $x$-coordinate is exactly halfway between the roots (or use $x = \frac{-b}{2a}$). Substitute $x$ back into the equation to find $y$.

Worked example 2 — Sketching a quadratic graph

Question: Sketch the graph of the quadratic function $y = x^2 - 2x - 3$.

1. Shape: $a = 1$ (positive), so it is a U-shape.

2. $y$-intercept: Set $x = 0$. $y = (0)^2 - 2(0) - 3$ $y = -3$ The $y$-intercept is at the point $(0, -3)$.

3. $x$-intercepts: Set $y = 0$. $x^2 - 2x - 3 = 0$ Factorize: $(x - 3)(x + 1) = 0$ $x - 3 = 0$ or $x + 1 = 0$ $x = 3$ or $x = -1$ The $x$-intercepts are at the points $(3, 0)$ and $(-1, 0)$.

4. Turning Point: The $x$-coordinate of the turning point is halfway between the roots $x = 3$ and $x = -1$. $x = \frac{3 + (-1)}{2} = \frac{2}{2} = 1$ Substitute $x = 1$ into the equation to find the $y$-coordinate: $y = (1)^2 - 2(1) - 3 = 1 - 2 - 3 = -4$ The turning point is at the point $(1, -4)$.

5. Sketch: Draw a U-shaped parabola crossing the $x$-axis at $-1$ and $3$, the $y$-axis at $-3$, with its lowest point at $(1, -4)$.

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Extended Content (Extended Curriculum Only)

(a) Cubic Graphs ($y = ax^3 + bx^2 + cx + d$)

Cubic graphs generally have an "S" or "N" shape and can have up to two turning points.

• Positive $x^3$: Starts bottom-left, ends top-right.
• Negative $x^3$: Starts top-left, ends bottom-right.

Worked example 3 — Sketching a cubic graph

Question: Sketch the graph of $y = (x-1)(x+1)(x-2)$.

1. $x$-intercepts: Set $y = 0$. $(x-1)(x+1)(x-2) = 0$ $x - 1 = 0$ or $x + 1 = 0$ or $x - 2 = 0$ $x = 1, x = -1, x = 2$ The $x$-intercepts are $(-1, 0)$, $(1, 0)$, and $(2, 0)$.

2. $y$-intercept: Set $x = 0$. $y = (0-1)(0+1)(0-2)$ $y = (-1)(1)(-2) = 2$ The $y$-intercept is $(0, 2)$.

3. Shape: Expanding the expression gives $x^3 + ...$ (positive $x^3$ term), so the graph starts from the bottom left and goes to the top right.

4. Sketch: Draw a smooth curve coming from the bottom left, passing through $x = -1$, turning, passing through $y = 2$, crossing down through $x = 1$, turning again, and crossing up through $x = 2$.

(b) Reciprocal Graphs ($y = \frac{k}{x}$)

These graphs occupy opposite quadrants and have asymptotes at the axes.

• $k > 0$: Curves in the 1st and 3rd quadrants.
• $k < 0$: Curves in the 2nd and 4th quadrants.

Key Feature: The graph never touches $x = 0$ or $y = 0$.

(c) Exponential Graphs ($y = a^x$)

Commonly $y = 2^x$ or $y = 3^x$.

• Always stays above the $x$-axis ($y > 0$).
• If $a > 1$, it shows growth.
• The $y$-intercept is always $(0, 1)$ because $a^0 = 1$.
• The $x$-axis ($y = 0$) is a horizontal asymptote.

Worked example 4 — Sketching an exponential graph

Question: Sketch the graph of $y = 3^x$.

1. $y$-intercept: Set $x = 0$. $y = 3^0 = 1$ The $y$-intercept is $(0, 1)$.

2. Asymptote: The $x$-axis ($y = 0$) is a horizontal asymptote. The graph approaches the $x$-axis as $x$ becomes increasingly negative, but never touches it.

3. Points: When $x = 1$, $y = 3^1 = 3$. The point is $(1, 3)$. When $x = 2$, $y = 3^2 = 9$. The point is $(2, 9)$.

4. Sketch: Draw a curve starting very close to the negative $x$-axis, rising slowly, passing through $(0, 1)$, and then curving sharply upwards, passing through $(1, 3)$ and $(2, 9)$.

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Key Equations

Linear: $y = mx + c$ ($m$ = gradient, $c$ = $y$-intercept)

Quadratic: $y = ax^2 + bx + c$ (Turning point at $x = \frac{-b}{2a}$)

Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ (Used to find $x$-intercepts (Roots) — provided on formula sheet)

Reciprocal: $y = \frac{k}{x}$ (Asymptotes at $x=0$ and $y=0$)

Exponential: $y = a^x$ ($y$-intercept at $(0, 1)$; $x$-axis is an asymptote)

Note: The Quadratic Formula is provided on the IGCSE formula sheet, but the method for finding the turning point or sketching linear/exponential graphs must be memorized.

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Common Mistakes to Avoid

• ❌ Wrong: Drawing a quadratic that looks like two straight lines joined at a point ("V" shape).
  • ✓ Right: Quadratics are smooth, curved parabolas. Never use a ruler to draw the curve.
• ❌ Wrong: Sketching a parabola without labeling the coordinates of the $x$ and $y$ intercepts.
  • ✓ Right: Always calculate and label the coordinates where the curve crosses the axes.
• ❌ Wrong: Drawing a reciprocal graph ($y = \frac{k}{x}$) or exponential graph ($y = a^x$) so that it touches its asymptote.
  • ✓ Right: Reciprocal and exponential graphs get very close to their asymptotes, but never intersect them. Leave a clear gap in your sketch.
• ❌ Wrong: Assuming that all quadratic graphs are U-shaped.
  • ✓ Right: Check the sign of the $x^2$ coefficient. If it's negative (e.g., $y = -x^2 + ...$), the parabola is an inverted "n" shape.

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Exam Tips

• Command Words:
  • "Sketch": No graph paper needed. Use a pencil. Show intercepts and general shape clearly.
  • "Plot": Use graph paper. Calculate a table of values and mark points precisely with an 'x'.
• Calculator vs. Non-Calculator: In a calculator paper, use the Table mode [Mode -> Table] to quickly find coordinates for tricky functions like cubics or exponentials.
• Check the $y$-intercept: This is the easiest mark. Always substitute $x = 0$ first.
• Symmetry: Remember that quadratic graphs are perfectly symmetrical. The turning point is always halfway between the roots.
• Real-world Context: If the graph represents a physical quantity (like time or distance), you may only need to sketch the part of the graph where $x \geq 0$.