Key features and models of graphs Problem set B Worked solutions

The function \(g(x)=-x^2+8x-7\) models a profit. Use your calculator to find the y-intercept, the zeros, the vertex and the equation of the axis of symmetry.

• The y-intercept is \(g(0)=-7\).
• From the calculator: zeros at \(x=1\) and \(x=7\)
• From the calculator: maximum at \((4,\,9)\)
• The axis of symmetry passes through the vertex: \(x=4\)

Write down the equation of the horizontal asymptote of each graph.

1. (a) \(y=2\times3^x-6\)

   • The constant is \(-6\): \(y=-6\)

2. (b) \(y=4\times0.8^x+1\)

   • The constant is 1: \(y=1\)

3. (c) \(y=3e^{-0.5x}-5\)

   • \(e^{-0.5x}\to0\), so \(y\to-5\): \(y=-5\)

Find the missing value in each case.

1. (a) y is directly proportional to \(x^2\), and \(y=48\) when \(x=4\). Find y when \(x=9\).

   • \(y=ax^2\): \(48=a(16)\), so \(a=3\)
   • \(y=3(9)^2=243\)

2. (b) y is inversely proportional to \(x^2\), and \(y=5\) when \(x=4\). Find the positive value of x for which \(y=20\).

   • \(y=\dfrac{a}{x^2}\): \(5=\dfrac{a}{16}\), so \(a=80\)
   • \(20=\dfrac{80}{x^2} \;\Rightarrow\; x^2=4\)
   • \(x=2\) (x is positive)

The function f is defined by \(f(x)=\dfrac{3x-12}{x+4}\), \(x\neq-4\). Write down the equations of the asymptotes and the coordinates of the intercepts with the axes.

• Vertical asymptote: \(x+4=0\), so \(x=-4\)
• Horizontal asymptote: \(y=\frac{3}{1}=3\)
• x-intercept: \(3x-12=0\), so \(x=4\)
• y-intercept: \(f(0)=\frac{-12}{4}=-3\)

In each case say whether a linear, quadratic or exponential model is most suitable, and give a reason.

1. (a) A car loses 12% of its value every year.

   • The value keeps 88% of itself each year, so the ratio is constant.

2. (b) The area of a rectangle of fixed perimeter, against its width.

   • The area is small for very thin and very wide rectangles and greatest in between: it rises, then falls.

3. (c) A plumber charges a fixed £30 call-out fee plus £45 for every hour.

   • A fixed call-out fee plus a constant amount per hour has a constant gradient.

A javelin follows the path \(h(x)=-0.04x^2+1.6x+1.8\), where x is the horizontal distance in metres and h is the height in metres.

1. (a) Find the greatest height of the javelin.

   • From the calculator: maximum at \((20,\,17.8)\)

2. (b) Find the horizontal distance at which the javelin reaches the ground.

   • From the calculator: zero at \(x=41.1\). The other zero is negative, so it is rejected.

3. (c) Say what the y-intercept tells you.

   • \(h(0)=1.8\)
   • The y-intercept is the height at which the javelin is released.

A metal block is taken from a furnace and left to cool. Its temperature in °C after t minutes is \(T(t)=21+179\times0.95^t\).

1. (a) Find the starting temperature of the block.

   • \(T(0)=21+179=200\)

2. (b) Write down the temperature that the block approaches. Say what it represents.

   • \(0.95^t\to0\), so \(T\to21\). The horizontal asymptote is \(T=21\): the temperature of the surroundings.

3. (c) Find the time for the block to cool to 60 °C. Give your answer to 3 significant figures.

   • Solve \(21+179\times0.95^t=60\)
   • From the calculator: \(t=29.7\)

The strength S of a radio signal, in units, is inversely proportional to the square of the distance d metres from the transmitter, so \(S=\dfrac{a}{d^2}\). At 3 m the strength is 32 units.

1. (a) Find a.

   • \(32=\dfrac{a}{9}\)
   • \(a=288\)

2. (b) Find the strength at 8 m.

   • \(S=\dfrac{288}{64}=4.5\)

3. (c) Explain what the vertical asymptote at \(d=0\) tells you about the model.

   • As d gets close to 0, S grows without limit.
   • The S-axis (\(d=0\)) is a vertical asymptote of the model.

Two sets of measurements are recorded.

1. (a) The height of a plant, in cm, after x weeks is 5, 8, 11, 14, 17 for x = 0, 1, 2, 3, 4. Choose a model, give a reason, write the model and predict the height at \(x=12\).

   • The differences are all 3, so the model is linear.
   • \(y=3x+5\)
   • \(y(12)=41\)

2. (b) The number of people who have heard a rumour after x hours is 3, 9, 27, 81 for x = 0, 1, 2, 3. Choose a model, give a reason, write the model and predict the number at \(x=6\).

   • The ratios are all 3, so the model is exponential.
   • \(y=3\times3^x\)
   • \(y(6)=2187\)

The function f is defined by \(f(x)=\dfrac{ax-5}{x+3}\), \(x\neq-3\). The horizontal asymptote of the graph is \(y=2\). Find a, then find the coordinates of the intercepts with the axes.

• The horizontal asymptote is \(y=a\), so \(a=2\).
• x-intercept: \(2x-5=0\), so \(x=2.5\)
• y-intercept: \(f(0)=\frac{-5}{3}=-1.67\)

The graph of \(y=ka^x+c\) has the horizontal asymptote \(y=-1\) and passes through \((0,\,1)\) and \((1,\,5)\). Find k, a and c, then find the value of y when \(x=4\).

• The asymptote gives \(c=-1\).
• At \(x=0\): \(k-1=1\), so \(k=2\).
• At \(x=1\): \(2a-1=5\), so \(a=3\).
• \(y=2\times3^4-1=161\)