Key features and models of graphs Problem set A Worked solutions

The function \(g(x)=-2x^2+12x-10\) 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 \(g(0)=-10\).
• From the calculator: zeros at \(x=1\) and \(x=5\)
• From the calculator: maximum at \((3,\,8)\)
• The axis of symmetry passes through the vertex: \(x=3\)

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

1. (a) \(y=3\times2^x-4\)

   • The constant is \(-4\): \(y=-4\)

2. (b) \(y=5\times0.5^x+2\)

   • The constant is 2: \(y=2\)

3. (c) \(y=-e^{-x}+7\)

   • \(e^{-x}\to0\), so \(y\to7\): \(y=7\)

Find the missing value in each case.

1. (a) y is directly proportional to \(x^2\), and \(y=45\) when \(x=3\). Find y when \(x=7\).

   • \(y=ax^2\): \(45=a(9)\), so \(a=5\)
   • \(y=5(7)^2=245\)

2. (b) y is inversely proportional to \(x^2\), and \(y=2\) when \(x=6\). Find the positive value of x for which \(y=8\).

   • \(y=\dfrac{a}{x^2}\): \(2=\dfrac{a}{36}\), so \(a=72\)
   • \(8=\dfrac{72}{x^2} \;\Rightarrow\; x^2=9\)
   • \(x=3\) (x is positive)

The function f is defined by \(f(x)=\dfrac{4x+8}{x-2}\), \(x\neq2\). Write down the equations of the asymptotes and the coordinates of the intercepts with the axes.

• Vertical asymptote: \(x-2=0\), so \(x=2\)
• Horizontal asymptote: \(y=\frac{4}{1}=4\)
• x-intercept: \(4x+8=0\), so \(x=-2\)
• y-intercept: \(f(0)=\frac{8}{-2}=-4\)

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

1. (a) The number of bacteria in a dish doubles every hour.

   • The number doubles each hour, so the ratio between values is constant.

2. (b) The height of a stone thrown upwards, against time.

   • A thrown stone goes up, reaches a maximum and comes down.

3. (c) A taxi charges a fixed £4 plus £1.50 for every kilometre.

   • A fixed charge plus a constant amount per kilometre has a constant gradient.

A basketball player's shot follows the path \(h(x)=-0.2x^2+1.2x+2\), where x is the horizontal distance in metres and h is the height in metres.

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

   • From the calculator: maximum at \((3,\,3.8)\)

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

   • From the calculator: zero at \(x=7.36\). The other zero is negative, so it is rejected.

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

   • \(h(0)=2\)
   • The y-intercept is the height at which the ball is released.

A hot drink cools in a room. Its temperature in °C after t minutes is \(T(t)=18+62\times0.9^t\).

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

   • \(T(0)=18+62=80\)

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

   • \(0.9^t\to0\), so \(T\to18\). The horizontal asymptote is \(T=18\): the temperature of the room.

3. (c) Find the time for the drink to cool to 25 °C. Give your answer to 3 significant figures.

   • Solve \(18+62\times0.9^t=25\)
   • From the calculator: \(t=20.7\)

The intensity I of a lamp's light, in lux, is inversely proportional to the square of the distance d metres from it, so \(I=\dfrac{a}{d^2}\). At 2 m the intensity is 45 lux.

1. (a) Find a.

   • \(45=\dfrac{a}{4}\)
   • \(a=180\)

2. (b) Find the intensity at 5 m.

   • \(I=\dfrac{180}{25}=7.2\)

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

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

Two sets of measurements are recorded.

1. (a) The depth of water in a tank, in cm, after x minutes is 3, 7, 11, 15, 19 for x = 0, 1, 2, 3, 4. Choose a model, give a reason, write the model and predict the depth at \(x=10\).

   • The differences are all 4, so the model is linear.
   • \(y=4x+3\)
   • \(y(10)=43\)

2. (b) The number of cells in a sample after x hours is 5, 10, 20, 40 for x = 0, 1, 2, 3. Choose a model, give a reason, write the model and predict the number at \(x=7\).

   • The ratios are all 2, so the model is exponential.
   • \(y=5\times2^x\)
   • \(y(7)=640\)

The function f is defined by \(f(x)=\dfrac{ax+3}{x-2}\), \(x\neq2\). The horizontal asymptote of the graph is \(y=4\). Find a, then find the coordinates of the intercepts with the axes.

• The horizontal asymptote is \(y=a\), so \(a=4\).
• x-intercept: \(4x+3=0\), so \(x=-0.75\)
• y-intercept: \(f(0)=\frac{3}{-2}=-1.5\)

The graph of \(y=ka^x+c\) has the horizontal asymptote \(y=3\) and passes through \((0,\,7)\) and \((1,\,11)\). Find k, a and c, then find the value of y when \(x=5\).

• The asymptote gives \(c=3\).
• At \(x=0\): \(k+3=7\), so \(k=4\).
• At \(x=1\): \(4a+3=11\), so \(a=2\).
• \(y=4\times2^5+3=131\)