Exercise Questions

All questions in this exercise are listed below. Click on a question to view its solution.

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Key Concepts

This exercise focuses on Function Transformations — techniques to sketch complex graphs by modifying a known "parent" function.

Vertical Translation

Shifting the graph up or down by adding or subtracting a constant outside the function: $y=f(x)+k$

• $k>0$: graph moves up by $k$ units
• $k<0$: graph moves down by $|k|$ units

Horizontal Translation

Shifting the graph left or right by adding or subtracting a constant inside the function argument: $y=f(x-h)$

• $h>0$: graph moves right by $h$ units
• $h<0$ (i.e., $f(x+h)$): graph moves left — this is counter-intuitive!

Vertical Stretching and Compression

Multiplying the function by a constant $a$: $y=a\cdot f(x)$

• $|a|>1$: vertical stretch (graph becomes steeper)
• $0<|a|<1$: vertical compression (graph becomes flatter)
• $a<0$: also causes a reflection across the $x$-axis

Horizontal Stretching and Compression

Multiplying the input $x$ by a constant $c$: $y=f(cx)$

• $|c|>1$: horizontal compression by factor $\frac{1}{c}$ (graph narrows)
• $0<|c|<1$: horizontal stretch by factor $\frac{1}{c}$ (graph widens)

Reflection

Flipping the graph across the $x$-axis: $y=-f(x)$ Every point $(x,y)$ maps to $(x,-y)$.

Parent Functions

Understanding the base shapes is essential before applying transformations:

• $f(x)=|x|$ — V-shaped, vertex at origin
• $f(x)=x^2$ — parabola, vertex at origin
• $f(x)=x^3$ — cubic, passes through origin
• $f(x)=\sqrt{x}$ — half-parabola, starts at origin

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Important Formulas

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Worked Example

Describe all transformations applied to $y=x^2$ to obtain $y=-2(x+3{)}^2+1$.

Step 1 — Identify the parent function: $f(x)=x^2$, vertex at $(0,0)$.

Step 2 — Horizontal shift: $(x+3)$ inside means shift left 3 units. New vertex: $(-3,0)$.

Step 3 — Vertical stretch and reflection: Multiply by $-2$. Since $|-2|=2>1$, there is a vertical stretch by factor 2. Since $-2<0$, there is also a reflection across the $x$-axis. New vertex: $(-3,0)$.

Step 4 — Vertical shift: $+1$ outside means shift up 1 unit. Final vertex: $(-3,1)$.

Result: The parabola opens downward, is stretched vertically by 2, and has vertex $(-3,1)$.

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Summary

This exercise explores Function Transformations, a powerful tool that allows mathematicians to sketch complex graphs by identifying a "parent" function and applying specific shifts or scales.

• Key Learnings: Students learn the counter-intuitive nature of horizontal shifts — $f(x+2)$ moves the graph left, not right.

• Strategy: Transform the anchor point (e.g., vertex $(0,0)$ for $x^2$) first, then apply scaling/reflection to remaining points.

• Verification: Use a graphical calculator to verify how changing a single constant physically moves the curve on the Cartesian plane.