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IB Mathematics: Applications and Interpretation HL · 12.5: Functions · 12.5.6

Name Date

Stretches and reflections Problem set A

Sketch by moving the key points: vertex, intercepts. Outside the bracket changes y, inside changes x. No calculator needed.

Difficulty: first skills applying them stretch

Skills

  1. 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.

    −2246−10−55xyy = f(x)
    1. (a) \(y=2f(x)\)
    2. (b) \(y=\frac{1}{3}f(x)\)

  2. 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.

    −3−2−112345−6−5−4−3−2−112xyy = f(x)
    1. (a) \(y=f(2x)\)
    2. (b) \(y=f(\frac{x}{3})\)

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

    1. (a) \(y=18x^2-12x\)
    2. (b) \(y=2x^2-\frac{4}{3}x\)
    3. (c) \(y=24x^2-8x\)

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

    1. (a) show that \(4f(x)=36x^2-24x\);
    2. (b) describe the transformation that maps \(y=f(x)\) onto \(y=36x^2-24x\).

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

    1. (a) \((3, −5)\)
    2. (b) \((−6, 2)\)

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

    1. (a) \((4, −1)\)
    2. (b) \((−3, −7)\)

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

    1. (a) \(y=-2f(x)\)
    2. (b) \(y=5f(2x)\)
    3. (c) \(y=-\frac{1}{2}f(x)\)
    4. (d) \(y=-3f(-2x)\)

In context

  1. 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.

    51015−22468101214xyy = f(x)
    1. (a) Arch 1 is \(y=\frac{3}{2}f(x)\). What is the greatest height, and where does the arch meet the ground?
    2. (b) Arch 2 is \(y=f(\frac{x}{2})\). Find the greatest height and the width of the arch at ground level.
    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 \(q\).

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

Stretch

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

  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.

jmaths.xyz · AI HL · 12.5.6 Stretches and reflections