Quadratic functions and their graphsIB MYP Maths Extended: Subtopic test

1. 1
   The graph of $y = x^2 - 4x - 5$ is drawn for values of $x$ from $-3$ to $7$.

   (a)

   Where does the graph cross the $y$-axis?

   [1 mark]

   • A$(0,5)$
   • B$(0,-4)$
   • C$(0,0)$
   • D$(0,-5)$

   (b)

   Where does the graph cross the $x$-axis?

   [1 mark]

   • A$x=-1$ and $x=5$
   • B$x=1$ and $x=-5$
   • C$x=-1$ and $x=-5$
   • D$x=4$ and $x=-5$

   (c)

   Find the equation of the axis of symmetry of the graph.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A footballer kicks a ball from the ground. Its height $h$ metres after $t$ seconds is given by $h = 20t - 5t^2$, for $0 \le t \le 4$.

   (a)

   At what time does the ball reach its greatest height?

   [1 mark]

   • A$4$ s
   • B$20$ s
   • C$2$ s
   • D$10$ s

   (b)

   The graph meets the $t$-axis again at $t=4$. What does this show?

   [1 mark]

   • AThe ball reaches its maximum height after $4$ seconds.
   • BThe ball lands on the ground after $4$ seconds.
   • CThe ball is kicked from a height of $4$ metres.
   • DThe ball is $4$ metres high after $1$ second.

   (c)

   Find the maximum height reached by the ball.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A function is defined by $f(x) = x^2 - 2x - 3$.

   (a)

   Find the coordinates of the points where the graph of $f$ crosses the axes.

   [3 marks]

   (b)

   Find the equation of the axis of symmetry and the coordinates of the vertex. State whether the vertex is a maximum or a minimum point.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A farmer uses $40$ m of fencing to enclose three sides of a rectangular vegetable plot; the fourth side is a wall. The two sides at right angles to the wall each have length $x$ metres, so the area of the plot is $A = x(40 - 2x)$ square metres.

   (a)

   (i) Find the values of $x$ for which $A=0$.
   (ii) Hence find the equation of the axis of symmetry of the graph of $A$ against $x$.
   (iii) Find the largest possible area of the plot.

   [6 marks]

   (b)

   Farmer B says that choosing $x=15$ must give a larger area than choosing $x=5$ because $15$ is bigger. Evaluate this claim, and explain why $x=25$ cannot be used.

   [6 marks]

   Total for question 4: 12 marks

End of questions