Quadratic & Cubic Graphs: Sketching, Roots, Turning Points & Applications

TAKEAWAYS:

• Quadratic graphs are U-shaped (parabolas) and can open upwards or downwards depending on the coefficient of $x^2$.
• Cubic graphs have one or two turning points and can cross the x-axis once, twice, or three times.
• The roots of a graph are the points where it crosses the x-axis (y = 0).
• Turning points (maximum or minimum) help determine the shape of the graph.
• Sketching graphs requires finding roots, y-intercept, and turning points before drawing.
• Real-world applications include profit-loss analysis, projectile motion, and optimization problems.

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1. Quadratic Graphs (Parabolas)

A quadratic equation is of the form: $$y=a{x}^2+bx+c$$ where $a\ne 0$.

Key Features of Quadratic Graphs

1. Shape: Always a parabola (U-shaped or upside-down U).
2. Direction:
   • If $a>0$, the parabola opens upwards.
   • If $a<0$, the parabola opens downwards.
3. Vertex (Turning Point): The highest or lowest point of the parabola.
4. Roots (x-intercepts): Points where the graph crosses the x-axis ($y=0$).
5. Y-intercept: Point where the graph crosses the y-axis ($x=0$).

How to Sketch a Quadratic Graph

1. Find the roots by solving $a{x}^2+bx+c=0$.
2. Find the y-intercept by setting $x=0$ (i.e., $y=c$).
3. Find the vertex using: $$x=-\frac{b}{2a}$$ Substitute this $x$ back into the equation to find $y$.
4. Plot the points and draw a smooth curve.

Example 1: Sketch $y=x^2-4x+3$

1. Find the roots: $$x^2-4x+3=0$$ $$(x-1)(x-3)=0$$ Roots: $x=1$ and $x=3$. Points: $(1,0)$ and $(3,0)$.

2. Find the y-intercept: Set $x=0$: $$y=0^2-4(0)+3=3$$ Point: $(0,3)$.

3. Find the vertex: $$x=-\frac{b}{2a}=-\frac{-4}{2(1)}=2$$ Substitute $x=2$ into the equation: $$y=(2{)}^2-4(2)+3=4-8+3=-1$$ Vertex: $(2,-1)$.

4. Sketch the graph:

   • Since $a=1>0$, the parabola opens upwards.
   • Plot the roots $(1,0)$ and $(3,0)$, y-intercept $(0,3)$, and vertex $(2,-1)$.
   • Draw a smooth curve through these points.

Comparison Table: Quadratic Graphs

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2. Cubic Graphs

A cubic equation is of the form: $$y=a{x}^3+b{x}^2+cx+d$$ where $a\ne 0$.

Key Features of Cubic Graphs

1. Shape: An S-shaped curve (can have one or two "bends").
2. Roots: Can have 1, 2, or 3 real roots (points where $y=0$).
3. Turning Points: Points where the graph changes direction (maximum or minimum).
   • A cubic graph can have one or two turning points.
4. Y-intercept: Found by setting $x=0$ (i.e., $y=d$).

How to Sketch a Cubic Graph

1. Find the roots by solving $a{x}^3+b{x}^2+cx+d=0$.
2. Find the y-intercept by setting $x=0$ (i.e., $y=d$).
3. Find the turning points by solving $\frac{dy}{dx}=0$ (derivative).
4. Plot the points and draw a smooth curve.

Example 2: Sketch $y=x^3-4x$

1. Find the roots: $$x^3-4x=0$$ $$x(x^2-4)=0$$ $$x=0,x=2,x=-2$$ Points: $(-2,0)$, $(0,0)$, $(2,0)$.

2. Find the y-intercept: Set $x=0$: $$y=0^3-4(0)=0$$ Point: $(0,0)$.

3. Find the turning points: Differentiate $y=x^3-4x$: $$\frac{dy}{dx}=3x^2-4$$ Set $\frac{dy}{dx}=0$: $$3x^2-4=0$$ $$x^2=\frac{4}{3}$$ $$x=\pm \frac{2\sqrt{3}}{3}\approx \pm 1.15$$ Substitute back to find $y$:

   • For $x=\frac{2\sqrt{3}}{3}$: $$y={\left(\frac{2\sqrt{3}}{3}\right)}^3-4\left(\frac{2\sqrt{3}}{3}\right)\approx -2.31$$
   • For $x=-\frac{2\sqrt{3}}{3}$: $$y={\left(-\frac{2\sqrt{3}}{3}\right)}^3-4\left(-\frac{2\sqrt{3}}{3}\right)\approx 2.31$$ Turning points: $\left(\frac{2\sqrt{3}}{3},-2.31\right)$ and $\left(-\frac{2\sqrt{3}}{3},2.31\right)$.

4. Sketch the graph:

   • Plot the roots $(-2,0)$, $(0,0)$, $(2,0)$.
   • Plot the turning points $\left(\frac{2\sqrt{3}}{3},-2.31\right)$ and $\left(-\frac{2\sqrt{3}}{3},2.31\right)$.
   • Draw a smooth S-shaped curve.

Comparison Table: Cubic Graphs

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3. Applications of Quadratic and Cubic Graphs

Quadratic Graphs in Real Life

1. Projectile Motion: The path of a thrown ball follows a parabola.
2. Profit-Loss Analysis: Businesses use quadratic graphs to find maximum profit or minimum loss.
3. Architecture: Parabolic shapes are used in bridges and arches for strength.

Cubic Graphs in Real Life

1. Population Growth: Models where growth slows down or speeds up.
2. Economics: Cost functions in production.
3. Engineering: Stress-strain relationships in materials.

Example 3: Profit-Loss Problem

A company’s profit $P$ (in thousands of rupees) depends on the number of items $x$ sold: $$P=-2x^2+50x-100$$ Find the maximum profit and the number of items sold at that profit.

1. Find the vertex (maximum profit since $a<0$): $$x=-\frac{b}{2a}=-\frac{50}{2(-2)}=12.5$$ Substitute $x=12.5$ into the equation: $$P=-2(12.5{)}^2+50(12.5)-100$$ $$P=-312.5+625-100=212.5$$ Maximum profit: 212.5 thousand rupees at 12.5 items.

2. Find the roots (break-even points): $$-2x^2+50x-100=0$$ $$2x^2-50x+100=0$$ $$x^2-25x+50=0$$ Using the quadratic formula: $$x=\frac{25\pm \sqrt{625-200}}{2}=\frac{25\pm \sqrt{425}}{2}$$ $$x\approx 20.5\text{ or }4.5$$ Break-even points: 4.5 items and 20.5 items.

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4. Solving Equations Graphically

Graphs can be used to solve equations by finding the points of intersection.

Example 4: Solve $x^2-3x+2=0$ Graphically

1. Graph the quadratic equation: $$y=x^2-3x+2$$ Find roots: $$x^2-3x+2=0$$ $$(x-1)(x-2)=0$$ Roots: $x=1$ and $x=2$.

2. Find the intersection with the x-axis: The roots are the solutions to the equation.

Example 5: Solve $x^3-4x=0$ Graphically

From the earlier cubic graph, the roots are: $$x=-2,0,2$$

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Exam Tip

1. Always label your graphs with equations, roots, y-intercepts, and turning points.
2. Show all steps when finding roots, y-intercepts, and turning points.
3. Practice sketching graphs from equations without a calculator.
4. Understand the shape of quadratic and cubic graphs based on the coefficients.
5. For SEE exams, expect questions on:
   • Sketching graphs from given equations.
   • Finding roots, y-intercepts, and turning points.
   • Solving equations graphically.
   • Real-world applications (e.g., profit-loss, projectile motion).

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SEE-Style Questions

Short Answer Questions

1. Sketch the graph of $y=-x^2+4x-3$ and find its vertex.
2. How many roots does the equation $y=x^3-x$ have? Sketch its graph.
3. What is the y-intercept of the graph $y=2x^3-5x^2+x-1$?

Long Answer Questions

4. A company’s profit $P$ (in thousands of rupees) is given by $P=-x^2+10x-12$, where $x$ is the number of units sold.

   • Sketch the graph of $P$.
   • Find the maximum profit and the number of units sold at that profit.
   • Determine the break-even points.

5. Sketch the graph of $y=x^3-3x^2$ and find:

   • The roots of the equation.
   • The coordinates of the turning points.
   • The y-intercept.

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Summary

• Quadratic graphs are parabolas with a vertex and roots.
• Cubic graphs are S-shaped with one or two turning points.
• Sketching graphs requires finding key points (roots, y-intercept, turning points).
• Real-world applications include profit analysis, projectile motion, and engineering.
• Graphical solutions help solve equations by finding intersections.

Practice sketching graphs daily to master this topic for your SEE exam!