L1 Mathematical Reasoning
Question Bank
Find the equation for each of the three parabolas shown in the diagram below. You must show your working.
The diagram below shows the graphs of $y = x^2 + 3x - 5$ and $y = \frac{1}{2}x + 1$.
Use the graph to estimate the solutions to the equation $x^2 + 3x - 5 = \frac{1}{2}x + 1$.
The graph of a parabola is shown below.
Use the graph to write down:
(i) the $x$-intercepts
(ii) the $y$-intercept
(iii) the coordinates of the vertex (turning point)
(iv) the equation of the axis of symmetry
(v) whether the vertex is a maximum or a minimum point.
Sketch the graph of $y = (x - 1)(x + 3)$.
On your sketch, label the $x$-intercepts, the $y$-intercept and the vertex with their coordinates.
A parabola has the equation $y = x^2 - 4x - 5$. Find the coordinates of its vertex.
Where would the graph of $y = 3x^2 - 5x - 2$ cut the x-axis?
The diagram below shows the graphs of a parabola, $y = 3x^2 - 4x + 1$, and a straight line, $y = x + 3$.
Use algebra to find the $x$-values of the two points where the graphs intersect.
A parabola has its vertex at $(3, -4)$ and also passes through the point $(5, 8)$.
Find the equation of the parabola in the form $y = ax^2 + bx + c$.
Find the equation for each of the two parabolas shown in the diagram below. You must show your working.
A graph has the equation $y = x^2 - 6x + 5$.
(i) Factorise the equation.
(ii) Describe THREE different features of the graph. Give coordinates where you can.
The diagram below shows the graphs of two parabolas, $y = 3x^2 + 10x - 1$ and $y = x^2 + 3x + 3$.
Use algebra to find the $x$-values of the two points of intersection for the graphs.
The main cable of a suspension bridge forms a parabola, with dimensions in metres as shown in the diagram below.
(i) By placing the vertex of the parabola at the origin $(0,0)$, find the equation that models the shape of the cable.
(ii) Find the height of the main cable above the roadway at a point 150 metres from the center of the bridge.
A logo is formed by two parabolas, similar to the one in the diagram. The vertices are at $(0, 4)$ and $(0, -4)$. The logo is 8 units wide at its widest point (the x-axis).
Find the equations of both the red (opening down) and blue (opening up) parabolas.
The graph of a parabola crosses the x-axis at $x = -2$ and $x = 6$. It also passes through the point $(1, -18)$.
Find the equation of the parabola in the form $y = ax^2 + bx + c$.
A rugby ball is kicked off a tee. Its height, $h$ metres, $t$ seconds after it is kicked is given by
$h = -5t^2 + 20t$
(i) After how many seconds does the ball hit the ground?
(ii) What is the greatest height the ball reaches?
(iii) For how many seconds is the ball higher than 15 m?
The parabola $y = x^2 + kx - 12$ passes through the point $(2, 0)$, where $k$ is a constant.
(i) Find the value of $k$.
(ii) Find the other $x$-intercept of the parabola.
(iii) Find the coordinates of the vertex.
The cross-section of a satellite dish is a parabola with its vertex at the origin. Its dimensions in metres are shown in the diagram below. The equation of the parabola is of the form $y = ax^2$.
Find the width of the dish at a point where its depth is exactly half of the maximum depth.
The parabola $y = 2x^2 + bx + c$ has its axis of symmetry at $x=3$. One of its x-intercepts is at $x=1$. Find the values of $b$ and $c$.
The diagram below shows the graph of the parabola $y = x^2 + bx + c$. The parabola passes through the two points marked on the graph. Use this information to find the values of $b$ and $c$.
The graph of the equation $y = ax^2 + c$ is a parabola that passes through the points $(2, 9)$ and $(4, 33)$. Find the values of the constants $a$ and $c$.
The entrance to a garden is a parabolic stone archway. The arch is 6 metres wide at the base and reaches a maximum height of 4.5 metres. A hook for a lantern is to be placed on the arch at a point that is 1 metre horizontally from the base on one side. Use algebra to find the height of the lantern hook above the ground.
The graph below shows the parabola P and the straight line L. They meet at the points A and B.
(i) Find the equation of the parabola P.
(ii) Find the equation of the line L.
(iii) Use an algebraic method to find the coordinates of A and B.
(iv) For what values of $x$ is the line L above the parabola P?
The entrance to a tunnel is a parabolic arch that is 8 metres wide at the base and has a maximum height of 5 metres. A rectangular truck is 3 metres wide and 4 metres high. Will the truck be able to pass through the tunnel? Justify your answer with algebraic working.
The cross-section of a valley is modelled by the parabola $h = k(x-150)^2 + 10$, where $h$ is the height in metres above sea level and $x$ is the horizontal distance in metres from a survey point. The valley is 200 metres wide at a height of 90 metres. A horizontal bridge is to be built across the valley at a height of 50 metres above sea level. Find the length of the bridge.
