Completing the square

The graph in Figure 3 shows the path of a small ball.

The ball travels in a vertical plane above horizontal ground.

The ball is thrown from the point represented by A and caught at the point represented by B.

The height, H metres, of the ball above the ground has been plotted against the horizontal distance, x metres, measured from the point where the ball was thrown.

With respect to a fixed origin O, the point A has coordinates (0, 2) and the point B has coordinates (20, 0.8), as shown in Figure 3.

The ball reaches its maximum height when x = 9

A quadratic function, linking H with x, is used to model the path of the ball.

Find H in terms of x.

Give one limitation of the model.

Chandra is standing directly under the path of the ball at a point 16 m horizontally from O.

Chandra can catch the ball if the ball is less than 2.5 m above the ground.

Use the model to determine if Chandra can catch the ball.

A circle C has equation

where k is a constant.

Find in terms of k,

the coordinates of the centre of C

the radius of C

The line with equation y = 2x – 1 intersects C at 2 distinct points.

Find the range of possible values of k.

Given that

express f (x) in the form (x + a)² + b where a and b are integers to be found. The curve with equation y = f (x)

• meets the y-axis at the point P

• has a minimum turning point at the point Q

Write down

the coordinates of P

the coordinates of Q

Given that

express f (x) in the form (x + a)² + b where a and b are integers to be found.

The curve with equation y = f (x)

• meets the y-axis at the point P

• has a minimum turning point at the point Q

Write down

the coordinates of P

the coordinates of Q

The circle C has equation

Find

the coordinates of the centre of C,

the exact radius of C, giving your answer as a simplified surd.

The line l has equation y = 3x + k where k is a constant.

Given that l is a tangent to C,

find the possible values of k, giving your answers as simplified surds.

Figure 3 is a graph of the trajectory of a golf ball after the ball has been hit until it first hits the ground.

The vertical height, H metres, of the ball above the ground has been plotted against the horizontal distance travelled, x metres, measured from where the ball was hit.

The ball is modelled as a particle travelling in a vertical plane above horizontal ground.

Given that the ball

• is hit from a point on the top of a platform of vertical height 3 m above the ground

• reaches its maximum vertical height after travelling a horizontal distance of 90 m

• is at a vertical height of 27 m above the ground after travelling a horizontal distance of 120 m

Given also that H is modelled as a quadratic function in x

find H in terms of x

Hence find, according to the model,

the maximum vertical height of the ball above the ground,

the horizontal distance travelled by the ball, from when it was hit to when it first hits the ground, giving your answer to the nearest metre.

The possible effects of wind or air resistance are two limitations of the model. Give one other limitation of this model.

f(x) = 2x² + 4x + 9 x

Write f(x) in the form a(x + b)² + c, where a, b and c are integers to be found.

Sketch the curve with equation y = f(x) showing any points of intersection with the coordinate axes and the coordinates of any turning point.

Describe fully the transformation that maps the curve with equation y = f(x) onto the curve with equation y = g(x) where

g(x) = 2(x – 2)² + 4x – 3 x

Find the range of the function

4x² + 8x + 3 ax + b² + c

Find the values of the constants a, b and c.

On the axes on page 27, sketch the curve with equation y = 4x² + 8x + 3, showing clearly the coordinates of any points where the curve crosses the coordinate axes.

4x – 5 – x² = q – (x + p)² where p and q are integers.

Find the value of p and the value of q.

Calculate the discriminant of 4x – 5 – x²

On the axes on page 17, sketch the curve with equation y = 4x – 5 – x² showing clearly the coordinates of any points where the curve crosses the coordinate axes.

Show that x² +6x + 11 can be written as

where p and q are integers to be found.

In the space at the top of page 7, sketch the curve with equation y x² 6x 11, showing clearly any intersections with the coordinate axes.

Find the value of the discriminant of x² +6x + 11

f(x) x² 4kx (311k), where k is a constant.

Express f(x) in the form (x + p)² + q, where p and q are constants to be found in terms of k.

Given that the equation f(x) = 0 has no real roots,

find the set of possible values of k.

Given that k = 1,

sketch the graph of y = f(x), showing the coordinates of any point at which the graph crosses a coordinate axis.