Stretches and reflections Problem set B Worked solutions

The graph of \(y=f(x)\) is shown. It has its maximum at (1, 12), crosses the x-axis at x = −1 and x = 3, and crosses the y-axis at (0, 9). Sketch each graph on its own axes, marking the vertex and the intercepts.

1. (a) \(y=3f(x)\)

   • Vertical stretch, scale factor 3: multiply every y-coordinate by 3; x-coordinates do not change.
   • Maximum: (1, 12) → (1, 36).
   • x-intercepts stay at −1 and 3 (y = 0 stays 0).
   • y-intercept: 9 × 3 = 27.

2. (b) \(y=\frac{1}{4}f(x)\)

   • Vertical stretch, scale factor 0.25: multiply every y-coordinate by 0.25; x-coordinates do not change.
   • Maximum: (1, 12) → (1, 3).
   • x-intercepts stay at −1 and 3 (y = 0 stays 0).
   • y-intercept: 9 × 0.25 = 2.25.

The graph of \(y=f(x)\) is shown. It has its minimum at (3, −4), crosses the x-axis at x = 1 and x = 5, and crosses the y-axis at (0, 5). Sketch each graph on its own axes, marking the vertex and the intercepts.

1. (a) \(y=f(3x)\)

   • Horizontal stretch, scale factor 1/3: divide every x-coordinate by 3; y-coordinates do not change.
   • Minimum: (3, −4) → (1, −4).
   • x-intercepts: 1 → 1/3 and 5 → 5/3.
   • y-intercept stays at (0, 5) (x = 0 stays 0).

2. (b) \(y=f(\frac{x}{2})\)

   • Horizontal stretch, scale factor 2: multiply every x-coordinate by 2; y-coordinates do not change.
   • Minimum: (3, −4) → (6, −4).
   • x-intercepts: 1 → 2 and 5 → 10.
   • y-intercept stays at (0, 5) (x = 0 stays 0).

Describe the transformation that maps the graph of \(y=4x^2-10x\) onto the graph with each equation.

1. (a) \(y=10x^2-25x\)

   • Let \(f(x)=4x^2-10x\).
   • \(10x^2-25x=\frac{5}{2}f(x)\): every term is multiplied by 2.5.
   • Vertical stretch, scale factor 2.5.

2. (b) \(y=2x^2-5x\)

   • Let \(f(x)=4x^2-10x\).
   • \(2x^2-5x=\frac{1}{2}f(x)\): every term is multiplied by 0.5.
   • Vertical stretch, scale factor 0.5.

3. (c) \(y=36x^2-30x\)

   • Let \(f(x)=4x^2-10x\).
   • Replace x by 3x: \(f(3x)=36x^2-30x\), which is the graph given.
   • Horizontal stretch, scale factor 1/3.

Given that \(f(x)=7x^2-14x\):

1. (a) show that \(3f(x)=21x^2-42x\);

   • \(3f(x)=3(7x^2-14x)\)
   • \(=21x^2-42x\)

2. (b) describe the transformation that maps \(y=f(x)\) onto \(y=21x^2-42x\).

   • The new equation is \(3f(x)\).
   • Vertical stretch, scale factor 3.

The graph of \(y=f(x)\) has a vertex at the given point. Find the vertex of \(y=-f(x)\).

1. (a) \((−4, 3)\)

   • \(y=-f(x)\) is a reflection in the x-axis: it multiplies each y-coordinate by −1.
   • (−4, 3) → (−4, −3)

2. (b) \((5, −8)\)

   • \(y=-f(x)\) is a reflection in the x-axis: it multiplies each y-coordinate by −1.
   • (5, −8) → (5, 8)

The graph of \(y=f(x)\) has a vertex at the given point. Find the vertex of \(y=f(-x)\).

1. (a) \((−8, 2)\)

   • \(y=f(-x)\) is a reflection in the y-axis: it multiplies each x-coordinate by −1.
   • (−8, 2) → (8, 2)

2. (b) \((7, −3)\)

   • \(y=f(-x)\) is a reflection in the y-axis: it multiplies each x-coordinate by −1.
   • (7, −3) → (−7, −3)

Describe each graph as a sequence of stretches and reflections of \(y=f(x)\).

1. (a) \(y=-4f(x)\)

   • Outside the bracket changes y; inside changes x.
   • Vertical stretch, scale factor 4.
   • Reflection in the x-axis.

2. (b) \(y=3f(4x)\)

   • Outside the bracket changes y; inside changes x.
   • Vertical stretch, scale factor 3.
   • Horizontal stretch, scale factor 0.25.

3. (c) \(y=-\frac{1}{3}f(x)\)

   • Outside the bracket changes y; inside changes x.
   • Vertical stretch, scale factor 1/3.
   • Reflection in the x-axis.

4. (d) \(y=-2f(-3x)\)

   • Outside the bracket changes y; inside changes x.
   • Vertical stretch, scale factor 2.
   • Reflection in the x-axis.
   • Horizontal stretch, scale factor 1/3.
   • Reflection in the y-axis.

The cross-section of a garden pond is modelled by \(y=f(x)\), where x and y are in metres and the water surface is the x-axis. The pond is 10 m wide and 2.5 m deep at its deepest point (5, −2.5). The graph is shown. A landscaper tests three other ponds.

1. (a) Pond 1 is \(y=\frac{8}{5}f(x)\). What is the greatest depth, and where does the pond meet the water surface?

   • \(y=\frac{8}{5}f(x)\) is a vertical stretch, scale factor 1.6: only the heights change.
   • Greatest depth: 2.5 × 1.6 = 4 m. The surface points x = 0 and 10 stay where they are.

2. (b) Pond 2 is \(y=f(\frac{x}{2})\). Find the greatest depth and the width of the pond at the surface.

   • \(y=f(\frac{x}{2})\) is a horizontal stretch, scale factor 2: only the widths change.
   • Depth stays 2.5 m. Surface points: x = 0 and x = 20, so the width is 20 m.

3. (c) Write down the equation of a pond with the same depth as the original but one third of the width, in the form \(y=f(qx)\), and give the value of \(q\).

   • A one-third-width pond needs a horizontal stretch, scale factor 1/3.
   • Scale factor 1/q = 1/3, so q = 3.

A survey of a river bed gives its profile as the graph of \(y=f(x)\), in metres, with the deepest point at (−5, −3). A model of the river is built as the mirror image \(y=f(-x)\), and a second model is the graph turned over, \(y=-f(x)\).

1. (a) Find the deepest point of the mirror-image model.

   • \(y=f(-x)\) reflects in the y-axis: multiply the x-coordinate by −1.
   • (−5, −3) → (5, −3)

2. (b) Find the highest point of the turned-over model.

   • \(y=-f(x)\) reflects in the x-axis: multiply the y-coordinate by −1.
   • (−5, −3) → (−5, 3)

Describe the transformation that maps the graph of \(y=4x^2-6x\) onto the graph of \(y=16x^2+12x\).

• Let \(f(x)=4x^2-6x\).
• Compare \(f(-2x)=16x^2+12x\) with the new equation.
• \(16x^2+12x=f(-2x)\).
• So: horizontal stretch, scale factor 0.5; reflection in the y-axis.

The point (8, 5) lies on the graph of \(y=f(x)\). The point (−4, 15) lies on the graph of \(y=pf(qx)\), where p and q are constants. Find p and q, and describe the transformation.

• A point \((a, b)\) on f moves to \(\left(\frac{a}{q}, pb\right)\).
• x: 8/q = −4, so q = −2.
• y: p × 5 = 15, so p = 3.
• Vertical stretch, scale factor 3; horizontal stretch, scale factor 0.5; reflection in the y-axis.