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 $x^2 + 3x - 5 = \frac{1}{2}x + 1$.
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.
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 $y = ax^2 + bx + c$.
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 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.
