Further Graphs And Tangents

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टीचर्स के लिए: Further Graphs And Tangents (Maths [CIE], Extended) के लिए इस्तेमाल के लिए तैयार लेसन स्लाइड्स, रिवीज़न नोट्स, डायग्राम — इन्हें अपने लेसन में इस्तेमाल करें, या टॉपिक को एक इंटरैक्टिव क्लास एक्टिविटी की तरह चलाएं जिसे आपके स्टूडेंट्स लाइव गेम की तरह खेलें।

लेसन नोट्स

Types of Graphs

Linear: y = mx + c or ax + by = c (straight line).
Quadratic: y = ax² + bx + c (parabola, symmetric).
Cubic: y = ax³ + bx² + cx (S-shaped curve).
Reciprocal: y = a/x + b (two branches, asymptotes at x=0 and y=b).
Exponential: y = a·kˣ + b (growth if k>1, decay if 0<k<1; asymptote at y=b).
Trigonometric: sine, cosine, tangent (periodic).

Types of graphs overview

Drawing Graphs from Tables

Substitute x-values into equation to get y-values; avoid x=0 for reciprocal graphs.
Plot points accurately (within half a square) and join with a smooth freehand curve.
Use calculator table function: enter function, start/end x, step size.
Check for symmetry (e.g., quadratics have vertical line of symmetry).
Be careful with negative numbers: use brackets and BIDMAS.

A quadratic plotted from a table of values

Solving Equations from Graphs

Solutions to f(x)=0 are x-intercepts (roots) of the graph.
To solve f(x)=k, draw horizontal line y=k and read intersection x-coordinates.
To solve f(x)=g(x), plot both graphs; intersection x-values are solutions.
Rearrange given equation to match the graph's equation plus a line.
Only give x-coordinates unless solving simultaneous equations (include y).

Finding the intersection of two graphs

Finding Gradients of Tangents

Gradient of curve at a point = gradient of tangent at that point.
Draw tangent by eye using a ruler; extend line for accuracy.
Calculate gradient = rise/run = Δy/Δx using two far-apart points on tangent.
Gradient represents rate of change (e.g., speed from distance-time graph).
Estimation is approximate; exact gradient requires differentiation.

Estimating a gradient from a tangent

Key Features of Reciprocal Graphs

y = a/x has vertical asymptote at x=0 and horizontal asymptote at y=0.
y = a/x + b shifts horizontal asymptote to y=b; vertical asymptote remains x=0.
y = 1/x² is always positive and steeper than y = 1/x.
No y-intercept or roots for y = a/x.

Asymptotes of y = 1/x

Key Features of Exponential Graphs

y = kˣ (k>1) shows exponential growth; y-intercept at (0,1); asymptote y=0.
y = kˣ (0<k<1) shows exponential decay; same intercept and asymptote.
y = a·kˣ + b has y-intercept at (0, a+b) and horizontal asymptote at y=b.
Negative powers (e.g., 2⁻ˣ) also represent decay.

Exponential growth

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साइन अप करें — फ्री लॉग इन

प्रैक्टिस सवाल

फ्री प्रीव्यू — 65 में से 8 सवाल। सभी देखने के लिए साइन अप करें।

1.Which of the following is the equation of a reciprocal graph?
Easy
- Ay = x²
- By = 1/x
- Cy = 2^x
- Dy = x³
2.What is the y-intercept of the graph y = 2^x?
Easy
- A(0, 0)
- B(0, 1)
- C(0, 2)
- D(1, 2)
3.Which type of graph has a vertical asymptote at x = 0?
Easy
- ALinear
- BQuadratic
- CReciprocal
- DExponential
4.The graph of y = x² is called a
Easy
- Astraight line
- Bparabola
- Ccubic curve
- Dhyperbola
5.Which of the following graphs represents exponential decay?
Easy
- Ay = 2^x
- By = (1/2)^x
- Cy = x²
- Dy = 1/x
6.The graph of y = x³ + 2x² - x - 2 is sketched. How many times does it cross the x-axis?
Medium
- A0
- B1
- C2
- D3
7.What is the horizontal asymptote of y = 3/x - 2?
Medium
- Ay = 0
- By = -2
- Cx = 0
- Dy = 3
8.The table shows values for y = 2x² - 3x + 1. Which y-value is missing? | x | 0 | 1 | 2 | |---|---|---|---| | y | 1 | 0 | ? |
Easy
- A1
- B3
- C5
- D7

और सवाल

9.To solve x² - 4x - 5 = 0 using the graph of y = x² - 4x - 5, you find the
Easy
- Ay-intercept
- Bturning point
- Cx-intercepts
- Dasymptotes
10.The gradient of the tangent to y = x² at x = 2 is estimated as 4. Which points on the tangent could give this gradient?
Medium
- A(1, 1) and (3, 9)
- B(0, -4) and (4, 12)
- C(2, 4) and (3, 8)
- D(1, 2) and (2, 6)
11.The graph of y = x² - 2x + 1 is drawn. To solve x² - 5x + 4 = 0, which line should be drawn?
Hard
- Ay = 3x - 3
- By = -3x + 3
- Cy = 3x + 3
- Dy = -3x - 3
12.The point (1, -6) lies on f(x) = x(x+2)(x-3). The tangent at this point has gradient -1. What is the y-intercept of the tangent?
Hard
- A-5
- B-7
- C-6
- D5
13.Using the graph of y = 2x³ - 4x² + 3, the equation 2x³ - 4x² + 3 = 1.5 has three solutions. How many solutions does 2x³ - 4x² + 3 = 5 have?
Medium
- A0
- B1
- C2
- D3
14.A curve has equation y = 20/x + x. It has a point where the gradient is -4, at (2, 12). What is the other point with gradient -4?
Hard
- A(-2, -12)
- B(4, 9)
- C(-4, -9)
- D(-2, 12)
15.Which equation represents the graph of a cubic function with a negative leading coefficient?
Medium
- Ay = x³ - 2x
- By = -x³ + 4x
- Cy = 2x² - 3
- Dy = -1/x
16.Which of the following is the graph of y = 1/x²?
Medium
- AAlways positive, two branches
- BAlways negative, two branches
- COne branch in first quadrant
- DOne branch in third quadrant
17.Which type of graph is represented by y = a/x + b?
Easy
- ALinear
- BQuadratic
- CReciprocal
- DExponential
18.What is the horizontal asymptote of y = 5/x + 3?
Easy
- Ay = 0
- By = 3
- Cy = 5
- Dx = 0
19.The graph of y = x² - 4x + 3 is drawn. Which line should be drawn to solve x² - 4x = 0?
Medium
- Ay = -3
- By = 0
- Cy = 3
- Dy = x
20.A tangent is drawn to the curve y = x³ at the point (1, 1). The tangent passes through (0, -2). What is the gradient of the tangent?
Medium
- A1
- B2
- C3
- D4
21.Which of the following is the graph of y = 2^x?
Easy
- AExponential growth
- BExponential decay
- CReciprocal
- DQuadratic
22.The table shows values for y = x² - 3x + 2. Which y-value is missing? | x | 0 | 1 | 2 | 3 | |---|---|---|---|---| | y | 2 | 0 | ? | 2 |
Easy
- A0
- B1
- C2
- D3
23.The graph of y = x³ - 4x is drawn. Use the graph to solve x³ - 4x = 0. How many solutions are there?
Medium
- A0
- B1
- C2
- D3
24.The graph of y = 1/x is shifted up by 4 units. What is the equation of the new graph?
Easy
- Ay = 1/(x+4)
- By = 1/x + 4
- Cy = 1/x - 4
- Dy = 4/x
25.What is the y-intercept of y = 3^x?
Easy
- A(0, 0)
- B(0, 1)
- C(0, 3)
- D(1, 0)
26.The curve y = x² - 2x + 1 is drawn. To solve x² - 5x + 4 = 0, which line should be drawn?
Medium
- Ay = 3x - 3
- By = -3x + 3
- Cy = 3x + 3
- Dy = -3x - 3
27.The tangent to y = x² at x = -1 has gradient -2. Which of the following points lies on this tangent?
Medium
- A(-1, 1)
- B(0, -1)
- C(1, 1)
- D(2, 4)
28.The graph of y = 2x³ - 4x² + 3 is drawn. The equation 2x³ - 4x² + 3 = k has only one solution for -1 ≤ x ≤ 2. Which integer value of k is possible?
Hard
- A0
- B2
- C4
- D6
29.The graph of y = x² - 4x - 2 is drawn. Use it to solve x² - 4x - 6 = 0 by drawing a suitable line. Which line?
Medium
- Ay = 4
- By = -4
- Cy = 2
- Dy = -2
30.The gradient of the tangent to y = x² at x = 2 is estimated as 4. Which pair of points on the tangent could give this gradient?
Easy
- A(1,1) and (3,9)
- B(1,0) and (3,8)
- C(2,4) and (4,12)
- D(0,0) and (2,4)
31.A curve has equation y = x² - 4x - 2. The graph is drawn. To solve x² - 4x - 6 = 0, which line should be drawn on the same axes?
Medium
- Ay = -4
- By = 4
- Cy = 2
- Dy = -2
32.The graph of y = x³ + x² - 3x - 1 is drawn. By drawing a suitable straight line, the equation x³ + x² - 4x = 0 can be solved. What is the equation of that line?
Medium
- Ay = x - 1
- By = x + 1
- Cy = -x - 1
- Dy = -x + 1
33.The graph of y = x² - 5x + 6 is drawn for 0 ≤ x ≤ 5. By drawing a suitable straight line, the solutions of x² - 5x = x - 7 can be found. What is the equation of this line?
Medium
- Ay = x - 7
- By = x - 1
- Cy = 2x - 1
- Dy = x + 1
34.The graph of y = 2x² - 4x - 1 is drawn. By drawing a suitable straight line, the solutions of x² - x - 1 = 0 can be found. What is the equation of this line?
Hard
- Ay = -2x + 1
- By = 2x - 1
- Cy = x + 1
- Dy = -x + 1
35.A curve has equation y = 20/x + x. It has a point where the gradient is -4, at (2, 12). What are the coordinates of the other point on the curve where the gradient is -4?
Hard
- A(-2, -12)
- B(-2, 12)
- C(2, -12)
- D(-12, -2)
36.The graph of y = 2x³ - 4x² + 3 is drawn for -1 ≤ x ≤ 2. The equation 2x³ - 4x² + 3 = k has only one solution for -1 ≤ x ≤ 2. Which integer value of k is possible?
Hard
- A2
- B-4
- C5
- D-2
37.The graph of y = 2^x is drawn. Which of the following points lies on the graph?
Easy
- A(3, 8)
- B(3, 6)
- C(2, 3)
- D(1, 1)
38.The graph of y = x³ + 3x² + 2 is drawn for -3.5 ≤ x ≤ 1.5. By drawing a suitable straight line, the equation x³ + 3x² + 2x + 2 = 0 can be solved. What is the equation of this line?
Medium
- Ay = -2x
- By = 2x
- Cy = -x
- Dy = x
39.The graph of y = x² - 4x - 2 is drawn. Use the graph to find estimates for the solutions of x² - 4x - 2 = 0. Which of the following is a correct pair of solutions?
Easy
- Ax ≈ -0.4 and x ≈ 4.4
- Bx ≈ 0.4 and x ≈ -4.4
- Cx ≈ 2.4 and x ≈ -2.4
- Dx ≈ 1.4 and x ≈ 3.4
40.The graph of y = x² - 4x - 2 is drawn. By drawing a suitable straight line on the same axes, the solutions of x² - 4x - 6 = 0 can be found. What is the equation of this line?
Medium
- Ay = 4
- By = -4
- Cy = 2
- Dy = -2
41.Match each graph type to its equation.
Easy
- Linear
- Quadratic
- Cubic
- Reciprocal
- y = ax² + bx + c
- y = mx + c
- y = a/x + b
- y = ax³ + bx² + cx
42.Arrange the following steps in the correct order to find the gradient of a curve at a point using a tangent.
Easy
- Draw a tangent to the curve at the point
- Find the difference in y and difference in x
- Choose two points on the tangent far apart
- Calculate gradient = rise/run
43.Which of the following are true about the graph of y = 2/x? (Select all that apply)
Easy
- AIt has a horizontal asymptote at y = 0
- BIt has a vertical asymptote at x = 0
- CIt passes through (0, 0)
- DIt is always positive
- EIt has no y-intercept
44.The graph of y = 3^x has a horizontal asymptote at y = 0.
Easy
True or false?
45.What is the y-intercept of the graph y = 2 × 3^x + 1?
Easy
- A(0, 3)
- B(0, 2)
- C(0, 1)
- D(0, 4)
46.Using the graph of y = x² - 4x - 2, which line should be drawn to solve x² - 4x - 6 = 0?
Medium
- Ay = 4
- By = -4
- Cy = 2
- Dy = -2
47.Which of the following are types of graphs you need to recognise for IGCSE Extended Maths? (Select all that apply)
Easy
- ALinear
- BQuadratic
- CCubic
- DLogarithmic
- EReciprocal
48.Which of the following is the graph of y = 1/x?
Easy
- AA hyperbola with two branches in quadrants I and III
- BA U-shaped curve
- CA straight line
- DA curve that increases exponentially
49.The graph of y = x² - 4x - 2 is drawn. Which line should be drawn to solve x² - 4x = 0?
Medium
- Ay = -2
- By = 2
- Cy = 0
- Dy = -4
50.Which of the following are true about the graph of y = 1/x²? (Select all that apply)
Medium
- AIt has a vertical asymptote at x = 0.
- BIt is always positive.
- CIt is steeper than y = 1/x.
- DIt has a horizontal asymptote at y = 0.
- EIt crosses the x-axis at x = 1.
51.Complete the table of values for y = 2x² - 3x + 1.
Easy
x: 0, 1, 2 y: 1, 0, ____
52.A tangent to the curve y = x³ at the point (1, 1) passes through (0, -2). Find the gradient of the tangent.
Medium
53.Complete the sentence: The graph of y = 1/x² is always ___ (positive/negative) and has a vertical asymptote at ___.
Easy
The graph of y = 1/x² is always ____ and has a vertical asymptote at ____.
54.Write down the equation of the horizontal asymptote of y = 5/x - 2.
Easy
55.A curve has equation y = 20/x + x. It has a point where the gradient is -4, at (2, 12). Find the other point on the graph where the gradient is -4. Give your answer as an ordered pair (x, y).
Hard
56.A curve has equation y = 2x³ - 5x + 1. Estimate the gradient of the curve at the point where x = 1 by drawing a tangent. Give your answer to 1 decimal place.
Medium
57.Complete the sentence: The graph of y = 1/x has a vertical asymptote at x = ___ and a horizontal asymptote at y = ___.
Easy
The graph of y = 1/x has a vertical asymptote at x = ____ and a horizontal asymptote at y = ____.
58.Write down the equation of the line that should be drawn on the graph of y = x² - 4x - 2 to solve x² - 4x - 6 = 0.
Medium
59.A curve has equation y = 20/x + x. The gradient at the point (2, 12) is -4. Find the x-coordinate of the other point on the curve where the gradient is -4.
Hard
60.Complete the table for y = 2x² - 3x + 1.
Easy
x: 0, 1, 2, 3 y: 1, 0, ____, ____
61.Describe how to estimate the gradient of a curve at a given point using a tangent. Include the formula you would use.
Medium
62.A tangent to the curve y = x³ at the point (1, 1) passes through (0, -2). Find the gradient of the tangent.
Medium
- A3
- B2
- C4
- D5
63.A curve has equation y = 20/x + x. It has a point where the gradient is -4, at (2, 12). Find the other point on the graph where the gradient is -4. Give your answer as an ordered pair (x, y).
Hard
- A(-2, -12)
- B(2, -12)
- C(-2, 12)
- D(-12, -2)
64.A curve has equation y = 2x³ - 5x + 1. Estimate the gradient of the curve at the point where x = 1 by drawing a tangent. Give your answer to 1 decimal place.
Medium
- A0
- B2
- C1
- D3
65.A curve has equation y = 20/x + x. The gradient at the point (2, 12) is -4. Find the x-coordinate of the other point on the curve where the gradient is -4.
Hard
- A2
- B-2
- C10
- D-10

रिवीज़न नोट्स

## Further Graphs & Tangents – Quick Revision ### Types of Graphs - **Linear**: \(y = mx + c\) or \(ax + by = c\) - **Quadratic**: \(y = ax^2 + bx + c\) (U-shaped, symmetrical) - **Cubic**: \(y = ax^3 + bx^2 + cx + d\) (S-shaped) - **Reciprocal**: \(y = \frac{a}{x}\) (two branches, asymptotes at \(x=0\) and \(y=0\)) - **Exponential**: \(y = a k^x + b\) (growth if \(k>1\), decay if \(0<k<1\); asymptote at \(y=b\)) - **Trigonometric**: sine, cosine, tangent (see Trigonometry notes) ### Drawing Graphs from Tables - Substitute \(x\) values into equation to get \(y\). - Plot points accurately (within half a square). - Join with a smooth freehand curve (no ruler for curves!). - Watch out: - For \(y = \frac{1}{x}\) or \(y = \frac{1}{x^2}\), \(x=0\) gives no value (vertical asymptote). - Use brackets for negative numbers, e.g. \(x=-3\) in \(y = -x^2 + 8x\): \(y = -(-3)^2 + 8(-3) = -9 -24 = -33\). ### Solving Equations from Graphs - **Intersection points**: Solve two equations simultaneously by plotting both graphs; the \(x\) (and \(y\)) coordinates of intersection are the solutions. - **Single equation**: Given graph of \(y=f(x)\), to solve \(f(x)=c\), draw horizontal line \(y=c\) and read \(x\) at intersections. - **Rearranging**: If given a different equation, rearrange to \(f(x) = mx + c\) and draw that line. - Only give \(x\)-coordinates unless asked for both. ### Finding Gradients of Tangents - Gradient of curve at a point = gradient of tangent at that point. - **To estimate**: 1. Draw tangent using a ruler (make it long). 2. Choose two points far apart on the tangent. 3. Gradient = \(\frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x}\). - The gradient represents rate of change (e.g. speed from distance-time, acceleration from speed-time). ### Key Tips - Reciprocal graphs: \(y = \frac{a}{x} + b\) shifts asymptote to \(y=b\). - Exponential: \(y = a k^x + b\) has asymptote \(y=b\), \(y\)-intercept \((0, a+b)\). - For cubic graphs, sketch by finding roots and \(y\)-intercept. - When drawing tangents, use a ruler and pick easy coordinates for gradient calculation.

Asymptote: A line that a curve approaches but never touches. Reciprocal graphs have vertical asymptote at x=0 and horizontal asymptote at y=0 (or y=b if shifted).
Tangent: A straight line that touches a curve at a point without crossing it. Its gradient equals the gradient of the curve at that point.
Gradient: The steepness of a line, calculated as rise/run (change in y divided by change in x). For a curve, it is estimated using the tangent.
Root (x-intercept): A point where the graph crosses the x-axis (y=0). Solutions to f(x)=0 are the x-coordinates of roots.
Point of intersection: A point where two graphs meet. Its coordinates satisfy both equations simultaneously.
Reciprocal graph: Graph of y = a/x (or y = a/x + b). Has two branches, no y-intercept, asymptotes at x=0 and y=0 (or y=b).
Exponential graph: Graph of y = a k^x + b. Shows growth (k>1) or decay (0<k<1). Has horizontal asymptote at y=b and y-intercept at (0, a+b).

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