Find intersection of exponential curves

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = 4 ^ { x }\) A copy of Figure 1, labelled Diagram 1, is shown on the next page.

1. On Diagram 1, sketch the curve with equation
   1. \(y = 2 ^ { x }\)
   2. \(y = 4 ^ { x } - 6\) Label clearly the coordinates of any points of intersection with the coordinate axes. The curve with equation \(y = 2 ^ { x }\) meets the curve with equation \(y = 4 ^ { x } - 6\) at the point \(P\).
2. Using algebra, find the exact coordinates of \(P\).
   \includegraphics[max width=\textwidth, alt={}]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-05_1009_1490_264_219}
   \section*{Diagram 1}

14. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the curves \(C _ { 1 }\) and \(C _ { 2 }\) $$\begin{aligned} & C _ { 1 } \text { has equation } y = 3 + \mathrm { e } ^ { x + 1 } \quad x \in \mathbb { R } \\ & C _ { 2 } \text { has equation } y = 10 - \mathrm { e } ^ { x } \quad x \in \mathbb { R } \end{aligned}$$ Given that \(C _ { 1 }\) and \(C _ { 2 }\) cut the \(y\)-axis at the points \(P\) and \(Q\) respectively,

1. find the exact distance \(P Q\). \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the point \(R\).
2. Find the exact coordinates of \(R\).

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5. (i) Describe fully a single transformation that maps the graph of \(y = 3 ^ { x }\) onto the graph of \(y = \left( \frac { 1 } { 3 } \right) ^ { x }\).
(ii) Sketch on the same diagram the curves \(y = \left( \frac { 1 } { 3 } \right) ^ { x }\) and \(y = 2 \left( 3 ^ { x } \right)\), showing the coordinates of any points where each curve crosses the coordinate axes. The curves \(y = \left( \frac { 1 } { 3 } \right) ^ { x }\) and \(y = 2 \left( 3 ^ { x } \right)\) intersect at the point \(P\).
(iii) Find the \(x\)-coordinate of \(P\) to 2 decimal places and show that the \(y\)-coordinate of \(P\) is \(\sqrt { 2 }\).

2. \includegraphics[max width=\textwidth, alt={}, center]{687756c0-2038-4077-8c5c-fe0ca0f6ce65-1_638_677_749_443} The diagram shows the curves \(y = 3 + 2 \mathrm { e } ^ { x }\) and \(y = \mathrm { e } ^ { x + 2 }\) which cross the \(y\)-axis at the points \(A\) and \(B\) respectively.

1. Write down the coordinates of \(A\) and \(B\). The two curves intersect at the point \(C\).
2. Find an expression for the \(x\)-coordinate of \(C\) and show that the \(y\)-coordinate of \(C\) is \(\frac { 3 \mathrm { e } ^ { 2 } } { \mathrm { e } ^ { 2 } - 2 }\).

9

1. Sketch the graph of \(y = 4 k ^ { x }\), where \(k\) is a constant such that \(k > 1\). State the coordinates of any points of intersection with the axes.
2. The point \(P\) on the curve \(y = 4 k ^ { x }\) has its \(y\)-coordinate equal to \(20 k ^ { 2 }\). Show that the \(x\)-coordinate of \(P\) may be written as \(2 + \log _ { k } 5\).
3. (a) Use the trapezium rule, with two strips each of width \(\frac { 1 } { 2 }\), to find an expression for the approximate value of $$\int _ { 0 } ^ { 1 } 4 k ^ { x } \mathrm {~d} x$$ (b) Given that this approximate value is equal to 16 , find the value of \(k\).

8 \includegraphics[max width=\textwidth, alt={}, center]{b2c1188d-881e-4fb5-bece-5a51006543c7-4_524_822_274_609} The diagram shows the curves \(y = a ^ { x }\) and \(y = 4 b ^ { x }\).

1. (a) State the coordinates of the point of intersection of \(y = a ^ { x }\) with the \(y\)-axis.
   (b) State the coordinates of the point of intersection of \(y = 4 b ^ { x }\) with the \(y\)-axis.
   (c) State a possible value for \(a\) and a possible value for \(b\).
2. It is now given that \(a b = 2\). Show that the \(x\)-coordinate of the point of intersection of \(y = a ^ { x }\) and \(y = 4 b ^ { x }\) can be written as $$x = \frac { 2 } { 2 \log _ { 2 } a - 1 } .$$

10. The figure 4 shows the curves \(\mathrm { f } ( x ) = A - B e ^ { - 0.5 x }\) and \(\mathrm { g } ( x ) = 26 + e ^ { 0.5 x }\) \begin{figure}[h] \end{figure} Given that \(\mathrm { f } ( x )\) passes through \(( 0,8 )\) and has an horizontal asymptote \(y = 48\) a. Find the values of \(A\) and \(B\) for \(\mathrm { f } ( x )\) (3)
b. State the range of \(\mathrm { g } ( x )\) (1) The curves \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) meet at the points \(C\) and \(D\) c. Find the \(x\)-coordinates of the intersection points \(C\) and \(D\), in the form \(\ln k\), where \(k\) is an integer.

5 \includegraphics[max width=\textwidth, alt={}, center]{8c0b68bd-2257-4994-b444-def0b3f64334-4_591_547_262_242} The diagram shows the graphs of \(y = 2 ^ { 3 x }\) and \(y = 2 ^ { 3 x + 2 }\). The graph of \(y = 2 ^ { 3 x }\) can be transformed to the graph of \(y = 2 ^ { 3 x + 2 }\) by means of a stretch.

1. Give details of the stretch. The point \(A\) lies on \(y = 2 ^ { 3 x }\) and the point \(B\) lies on \(y = 2 ^ { 3 x + 2 }\). The line segment \(A B\) is parallel to the \(y\)-axis and the difference between the \(y\)-coordinates of \(A\) and \(B\) is 36 .
2. Determine the \(x\)-coordinate of \(A\). Give your answer in the form \(m \log _ { 2 } n\) where \(m\) and \(n\) are constants to be determined.

8 The diagram shows a sketch of the curve with equation \(y = 3 ^ { x } + 1\). \includegraphics[max width=\textwidth, alt={}, center]{4a4d4dcd-4137-427d-834f-ac2fe83f8aeb-5_535_1011_411_513} The curve intersects the \(y\)-axis at the point \(A\).

1. Write down the \(y\)-coordinate of point \(A\).
2. 1. Use the trapezium rule with five ordinates (four strips) to find an approximation for \(\int _ { 0 } ^ { 1 } \left( 3 ^ { x } + 1 \right) \mathrm { d } x\), giving your answer to three significant figures.
      (4 marks)
   2. By considering the graph of \(y = 3 ^ { x } + 1\), explain with the aid of a diagram whether your approximation will be an overestimate or an underestimate of the true value of \(\int _ { 0 } ^ { 1 } \left( 3 ^ { x } + 1 \right) \mathrm { d } x\).
      (2 marks)
3. The line \(y = 5\) intersects the curve \(y = 3 ^ { x } + 1\) at the point \(P\). By solving a suitable equation, find the \(x\)-coordinate of the point \(P\). Give your answer to four decimal places.
   (4 marks)
4. The curve \(y = 3 ^ { x } + 1\) is reflected in the \(y\)-axis to give the curve with equation \(y = \mathrm { f } ( x )\). Write down an expression for \(\mathrm { f } ( x )\).
   (1 mark)

7

1. Describe a geometrical transformation that maps the graph of \(y = 4 ^ { x }\) onto the graph of \(y = 3 \times 4 ^ { x }\).
2. Sketch the curve with equation \(y = 3 \times 4 ^ { x }\), indicating the value of the intercept on the \(y\)-axis.
3. The curve with equation \(y = 4 ^ { - x }\) intersects the curve \(y = 3 \times 4 ^ { x }\) at the point \(P\). Use logarithms to find the \(x\)-coordinate of \(P\), giving your answer to three significant figures.

8 The diagram shows a sketch of the curve with equation \(y = 6 ^ { x }\). \includegraphics[max width=\textwidth, alt={}, center]{a2525df8-dbd0-4b69-b6bb-f8ef6f96f7dc-5_403_506_370_769}

1. 1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for \(\int _ { 0 } ^ { 2 } 6 ^ { x } \mathrm {~d} x\), giving your answer to three significant figures.
   2. Explain, with the aid of a diagram, whether your approximate value will be an overestimate or an underestimate of the true value of \(\int _ { 0 } ^ { 2 } 6 ^ { x } \mathrm {~d} x\).
2. 1. Describe a single geometrical transformation that maps the graph of \(y = 6 ^ { x }\) onto the graph of \(y = 6 ^ { 3 x }\).
   2. The line \(y = 84\) intersects the curve \(y = 6 ^ { 3 x }\) at the point \(A\). By using logarithms, find the \(x\)-coordinate of \(A\), giving your answer to three decimal places.
      (4 marks)
3. The graph of \(y = 6 ^ { x }\) is translated by \(\left[ \begin{array} { c } 1 \\ - 2 \end{array} \right]\) to give the graph of the curve with equation \(y = \mathrm { f } ( x )\). Write down an expression for \(\mathrm { f } ( x )\).

8

1. Sketch the curve with equation \(y = 7 ^ { x }\), indicating the coordinates of any point where the curve intersects the coordinate axes.
2. The curve \(C _ { 1 }\) has equation \(y = 7 ^ { x }\). The curve \(C _ { 2 }\) has equation \(y = 7 ^ { 2 x } - 12\).
   1. By forming and solving a quadratic equation, prove that the curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at exactly one point. State the \(y\)-coordinate of this point.
   2. Use logarithms to find the \(x\)-coordinate of the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\), giving your answer to three significant figures.
      (2 marks)

5. (a) Describe fully a single transformation that maps the graph of \(y = 3 ^ { x }\) onto the graph of \(y = \left( \frac { 1 } { 3 } \right) ^ { x }\).
(b) Sketch on the same diagram the curves \(y = \left( \frac { 1 } { 3 } \right) ^ { x }\) and \(y = 2 \left( 3 ^ { x } \right)\), showing the coordinates of any points where each curve crosses the coordinate axes. The curves \(y = \left( \frac { 1 } { 3 } \right) ^ { x }\) and \(y = 2 \left( 3 ^ { x } \right)\) intersect at the point \(P\).
(c) Find the \(x\)-coordinate of \(P\) to 2 decimal places and show that the \(y\)-coordinate of \(P\) is \(\sqrt { 2 }\).

2. \begin{figure}[h] \end{figure} Figure 1 shows the curves \(y = 3 + 2 \mathrm { e } ^ { x }\) and \(y = \mathrm { e } ^ { x + 2 }\) which cross the \(y\)-axis at the points \(A\) and \(B\) respectively.

1. Find the exact length \(A B\). The two curves intersect at the point \(C\).
2. Find an expression for the \(x\)-coordinate of \(C\) and show that the \(y\)-coordinate of \(C\) is \(\frac { 3 \mathrm { e } ^ { 2 } } { \mathrm { e } ^ { 2 } - 2 }\).

6 The diagram shows a sketch of the curve with equation \(y = 3 \left( 2 ^ { x } + 1 \right)\). \includegraphics[max width=\textwidth, alt={}, center]{ad574bde-3bf1-45be-a454-9c723088b357-5_465_851_390_607} The curve \(y = 3 \left( 2 ^ { x } + 1 \right)\) intersects the \(y\)-axis at the point \(A\).

1. Find the \(y\)-coordinate of the point \(A\).
2. Use the trapezium rule with four ordinates (three strips) to find an approximate value for \(\int _ { 0 } ^ { 6 } 3 \left( 2 ^ { x } + 1 \right) d x\).
3. The line \(y = 21\) intersects the curve \(y = 3 \left( 2 ^ { x } + 1 \right)\) at the point \(P\).
   1. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$2 ^ { x } = 6$$
   2. Use logarithms to find the \(x\)-coordinate of \(P\), giving your answer to three significant figures.