Stretches and reflections Problem set A One question at a time

The graph of \(y=f(x)\) is shown. It has its maximum at (3, 6), crosses the x-axis at x = 1 and x = 5, and crosses the y-axis at (0, −7.5). Sketch each graph on its own axes, marking the vertex and the intercepts.

1. (a) \(y=2f(x)\)

   • Vertical stretch, scale factor 2: multiply every y-coordinate by 2; x-coordinates do not change.
   • Maximum: (3, 6) → (3, 12).
   • x-intercepts stay at 1 and 5 (y = 0 stays 0).
   • y-intercept: −7.5 × 2 = −15.

2. (b) \(y=\frac{1}{3}f(x)\)

   • Vertical stretch, scale factor 1/3: multiply every y-coordinate by 1/3; x-coordinates do not change.
   • Maximum: (3, 6) → (3, 2).
   • x-intercepts stay at 1 and 5 (y = 0 stays 0).
   • y-intercept: −7.5 × 1/3 = −2.5.

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

1. (a) \(y=f(2x)\)

   • Horizontal stretch, scale factor 0.5: divide every x-coordinate by 2; y-coordinates do not change.
   • Minimum: (1, −4) → (0.5, −4).
   • x-intercepts: −1 → −0.5 and 3 → 1.5.
   • y-intercept stays at (0, −3) (x = 0 stays 0).

2. (b) \(y=f(\frac{x}{3})\)

   • Horizontal stretch, scale factor 3: multiply every x-coordinate by 3; y-coordinates do not change.
   • Minimum: (1, −4) → (3, −4).
   • x-intercepts: −1 → −3 and 3 → 9.
   • y-intercept stays at (0, −3) (x = 0 stays 0).

Describe the transformation that maps the graph of \(y=6x^2-4x\) onto the graph with each equation.

1. (a) \(y=18x^2-12x\)

   • Let \(f(x)=6x^2-4x\).
   • \(18x^2-12x=3f(x)\): every term is multiplied by 3.
   • Vertical stretch, scale factor 3.

2. (b) \(y=2x^2-\frac{4}{3}x\)

   • Let \(f(x)=6x^2-4x\).
   • \(2x^2-\frac{4}{3}x=\frac{1}{3}f(x)\): every term is multiplied by 1/3.
   • Vertical stretch, scale factor 1/3.

3. (c) \(y=24x^2-8x\)

   • Let \(f(x)=6x^2-4x\).
   • Replace x by 2x: \(f(2x)=24x^2-8x\), which is the graph given.
   • Horizontal stretch, scale factor 1/2.

Given that \(f(x)=9x^2-6x\):

1. (a) show that \(4f(x)=36x^2-24x\);

   • \(4f(x)=4(9x^2-6x)\)
   • \(=36x^2-24x\)

2. (b) describe the transformation that maps \(y=f(x)\) onto \(y=36x^2-24x\).

   • The new equation is \(4f(x)\).
   • Vertical stretch, scale factor 4.

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

1. (a) \((3, −5)\)

   • \(y=-f(x)\) is a reflection in the x-axis: it multiplies each y-coordinate by −1.
   • (3, −5) → (3, 5)

2. (b) \((−6, 2)\)

   • \(y=-f(x)\) is a reflection in the x-axis: it multiplies each y-coordinate by −1.
   • (−6, 2) → (−6, −2)

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

1. (a) \((4, −1)\)

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

2. (b) \((−3, −7)\)

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

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

1. (a) \(y=-2f(x)\)

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

2. (b) \(y=5f(2x)\)

   • Outside the bracket changes y; inside changes x.
   • Vertical stretch, scale factor 5.
   • Horizontal stretch, scale factor 0.5.

3. (c) \(y=-\frac{1}{2}f(x)\)

   • Outside the bracket changes y; inside changes x.
   • Vertical stretch, scale factor 0.5.
   • Reflection in the x-axis.

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

   • Outside the bracket changes y; inside changes x.
   • Vertical stretch, scale factor 3.
   • Reflection in the x-axis.
   • Horizontal stretch, scale factor 0.5.
   • Reflection in the y-axis.

The cross-section of a footbridge arch is modelled by \(y=f(x)\), where x and y are in metres and the ground is the x-axis. The arch is 8 m wide and 8 m high at its highest point (4, 8). The graph is shown. Engineers test three other arches.

1. (a) Arch 1 is \(y=\frac{3}{2}f(x)\). What is the greatest height, and where does the arch meet the ground?

   • \(y=\frac{3}{2}f(x)\) is a vertical stretch, scale factor 1.5: only the heights change.
   • Greatest height: 8 × 1.5 = 12 m. The ground points x = 0 and 8 stay where they are.

2. (b) Arch 2 is \(y=f(\frac{x}{2})\). Find the greatest height and the width of the arch at ground level.

   • \(y=f(\frac{x}{2})\) is a horizontal stretch, scale factor 2: only the widths change.
   • Height stays 8 m. Ground points: x = 0 and x = 16, so the width is 16 m.

3. (c) Write down the equation of an arch with the same height as the original but half the width, in the form \(y=f(qx)\), and give the value of \(q\).

   • A half-width arch needs a horizontal stretch, scale factor 0.5.
   • Scale factor 1/q = 0.5, so q = 2.

A designer draws the profile of a roof as the graph of \(y=f(x)\), in metres. Its highest point is at (6, 4). She draws a second roof as \(y=f(-x)\) and a third by turning the first upside down, \(y=-f(x)\).

1. (a) Find the highest point of the second roof.

   • \(y=f(-x)\) reflects in the y-axis: multiply the x-coordinate by −1.
   • (6, 4) → (−6, 4)

2. (b) Find the lowest point of the third roof.

   • \(y=-f(x)\) reflects in the x-axis: multiply the y-coordinate by −1.
   • (6, 4) → (6, −4)

Describe the transformation that maps the graph of \(y=6x^2-4x\) onto the graph of \(y=-1.5x^2+2x\).

• Let \(f(x)=6x^2-4x\).
• Compare \(f(\frac{x}{2})=1.5x^2-2x\) with the new equation.
• \(-1.5x^2+2x=-f(\frac{x}{2})\).
• So: reflection in the x-axis; horizontal stretch, scale factor 2.

The point (6, −4) lies on the graph of \(y=f(x)\). The point (2, 12) 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: 6/q = 2, so q = 3.
• y: p × −4 = 12, so p = −3.
• Vertical stretch, scale factor 3; reflection in the x-axis; horizontal stretch, scale factor 1/3.