Graphs & curve sketching

Figure 3 shows a sketch of the graph with equation y = f (x) where f (x) = 4 x – 3 – 5 x Given that a is a constant and a = 1

find the possible values of f (a) The function g is defined by g (x) = 2x + 17 x

Find the range of g f (x) The function h is defined by h (x) = kx x where k is a constant. Given that the equation f (x) = h (x) has no solutions,

find the range of values of k.

Figure 1 shows a sketch of the graph with equation

The vertex of the graph is at the point P, shown in Figure 1.

Find the coordinates of P.

Solve the equation

A line l has equation y = kx + 4 where k is a constant.

Given that l intersectsy 3 x 2 5at 2 distinct points,

find the range of values of k.

The number of subscribers to two different music streaming companies is being monitored. The number of subscribers, N_A, in thousands, to company A is modelled by the equation

where t is the time in years since monitoring began. The number of subscribers, N_B, in thousands, to company B is modelled by the equation

where t is the time in years since monitoring began. Figure 2 shows a sketch of the graph of N_A and the graph of N_B over a 5-year period. Use the equations of the models to answer parts (a), (b), (c) and (d).

Find the initial difference between the number of subscribers to company A and the number of subscribers to company B. When t = T company A reduced its subscription prices and the number of subscribers increased.

Suggest a value for T, giving a reason for your answer.

Find the range of values of t for which N_A N_B giving your answer in set notation.

State a limitation of the model used for company B.

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

Figure 1 shows a sketch of the graph with equation y = |3 − 2x| Solve | 3 − 2 x | = 7 + x

Figure 4 shows a sketch of the graph with equation

y = | 2 x – 3 k |

where k is a positive constant.

Sketch the graph with equation y = f (x) where

f (x) = k – | 2 x – 3 k |

stating

• the coordinates of the maximum point

• the coordinates of any points where the graph cuts the coordinate axes

Find, in terms of k, the set of values of x for which

giving your answer in set notation.

Find, in terms of k, the coordinates of the minimum point of the graph with equation

Figure 4 shows a sketch of the graph with equation y = | 2 x – 3 k | where k is a positive constant.

Sketch the graph with equation y = f (x) where f (x) = k – | 2 x – 3 k | stating

• the coordinates of the maximum point

• the coordinates of any points where the graph cuts the coordinate axes

Find, in terms of k, the set of values of x for which

giving your answer in set notation.

Find, in terms of k, the coordinates of the minimum point of the graph with equation

Figure 2 shows a sketch of the graph with equation y = 2 | x + 4 | − 5 The vertex of the graph is at the point P, shown in Figure 2.

Find the coordinates of P.

Solve the equation 3 x + 40 = 2 | x + 4 | − 5

A line l has equation y = ax, where a is a constant. Given that l intersects y = 2| x + 4| − 5 at least once,

find the range of possible values of a, writing your answer in set notation.

Figure 1 shows a plot of part of the curve with equation y = cos x where x is measured in radians. Diagram 1, on the opposite page, is a copy of Figure 1.

Use Diagram 1 to show why the equation

has only one real root, giving a reason for your answer.

Given that the root of the equation is , and that is small,

use the small angle approximation for cos x to estimate the value of to 3 decimal places.

f(x) = 10e^–0.25x sin x, x 0

Show that the x coordinates of the turning points of the curve with equation y = f(x) satisfy the equation tan x = 4

Figure 3 shows a sketch of part of the curve with equation y = f(x).

Sketch the graph of H against t where

showing the long-term behaviour of this curve.

The function H(t) is used to model the height, in metres, of a ball above the ground t seconds after it has been kicked.

Using this model, find

the maximum height of the ball above the ground between the first and second bounce.

Explain why this model should not be used to predict the time of each bounce.

Explain why you can deduce that q = 4

Figure 4 shows a sketch of part of the curve C. The region R, shown shaded in Figure 4, is bounded by the curve C, the x-axis and the line with equation x = 3

Show that the exact value of the area of R is a ln 2 + b ln 3, where a and b are rational constants to be found.

A curve C has parametric equations

Show that all points on C satisfy y 6(x 3)²

Sketch the curve C.

Explain briefly why C does not include all points of y 6(x 3)², x

The line with equation x y k, where k is a constant, intersects C at two distinct points.

State the range of values of k, writing your answer in set notation.

“If m and n are irrational numbers, where m n, then mn is also irrational.”

Disprove this statement by means of a counter example.

Sketch the graph of y x 3

Explain why x 3 x 3 for all real values of x.

Factorise completely 9x – 4x³

Sketch the curve C with equation

Show on your sketch the coordinates at which the curve meets the x-axis.

The points A and B lie on C and have x coordinates of –2 and 1 respectively.

Show that the length of AB is k 10 where k is a constant to be found.

The curve C passes through the point (—1, 0) and touches the x-axis at the point (2, 0). The curve C has a maximum at the point (0, 4).

The equation of the curve C can be written in the form

where a, b andc are integers. Calculate the values of a, b and c. (5)

Show clearly the coordinates of all the points where the curve crosses or meets the coordinate axes. (3)

Figure 1 shows a sketch of the curve with equation y = , x 0 The curve C has equation y = − 5, x 0, and the line l has equation y = 4x + 2

Sketch and clearly label the graphs of C and l on a single diagram. On your diagram, show clearly the coordinates of the points where C and l cross the coordinate axes.

Write down the equations of the asymptotes of the curve C.

Find the coordinates of the points of intersection of y = − 5 and y = 4x + 2

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

On the axes below sketch the graphs of

y = x (4 – x)

y = x² (7 – x) showing clearly the coordinates of the points where the curves cross the coordinate axes.

Show that the x-coordinates of the points of intersection of

are given by the solutions to the equation x(x² – 8x + 4) = 0

The point A lies on both of the curves and the x and y coordinates of A are both positive.

Find the exact coordinates of A, leaving your answer in the form (p q√3, r + s√3), where p, q, r and s are integers.

Factorise completely x³ 4x

Sketch the curve C with equation

showing the coordinates of the points at which the curve meets the x-axis.

The point A with x-coordinate 1 and the point B with x-coordinate 3 lie on the curve C.

Find an equation of the line which passes through A and B, giving your answer in the form y = mx + c, where m and c are constants.

Show that the length of AB is k 10, where k is a constant to be found.

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.

Factorise completely x³ – 6x² + 9x

Sketch the curve with equation

showing the coordinates of the points at which the curve meets the x-axis.

Using your answer to part (b), or otherwise,

sketch, on a separate diagram, the curve with equation

showing the coordinates of the points at which the curve meets the x-axis.

The point P (1, a) lies on the curve with equation y = (x + 1)²(2 – x).

Find the value of a.

On the axes below sketch the curves with the following equations:

On your diagram show clearly the coordinates of any points at which the curves meet the axes.

With reference to your diagram in part (b) state the number of real solutions to the equation

The curve C has equation y=y = and the line l has equation y = 2x + 5.

On the axes below, sketch the graphs of C and l, indicating clearly the coordinates of any intersections with the axes.

Find the coordinates of the points of intersection of C and l.

The curve C has equation

Sketch C showing clearly the coordinates of the points where the curve meets the coordinate axes.

Show that the equation of C can be written in the form

where k is a positive integer, and state the value of k.

There are two points on C where the gradient of the tangent to C is equal to 3.

Find the x-coordinates of these two points.