Further Graphs

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टीचर्स के लिए: Further Graphs (Maths [CIE], Core) 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

Straight lines: y = mx + c (e.g., y = 3x + 2). Key lines: y = x and y = -x.
Horizontal lines: y = c (e.g., y = 4).
Vertical lines: x = k (e.g., x = 2).
Quadratic graphs: y = ax² + bx + c. Shape is a parabola: u-shaped (positive a) or n-shaped (negative a).
Reciprocal graphs: y = a/x. Two L-shaped branches; x ≠ 0. Positive a gives branches in first and third quadrants.

Positive and negative quadratics

Key Features of Quadratic Graphs

The turning point is called the vertex: minimum for positive quadratics, maximum for negative quadratics.
Quadratic graphs have a vertical line of symmetry through the vertex: x = k (k is x-coordinate of vertex).
Roots are x-intercepts where y = 0. A quadratic can have 0, 1 (touches), or 2 roots.
Roots are symmetric about the line of symmetry.
Quadratic graphs always have one y-intercept (where x = 0).

Key features of a quadratic graph

Drawing Graphs from Tables

Substitute x-values into the equation to find y-values. Use brackets for negative x and follow BIDMAS.
For reciprocal graphs, do not include x = 0 (division by zero).
Plot points accurately (within half a square). Join with a smooth freehand curve (no ruler).
Use calculator table function: enter function, start/end x, step size. Check given y-values.
If a point doesn't fit the curve shape, check your working.

Solving Equations from Graphs

To solve f(x) = 0, read the x-intercepts (roots) of the graph of y = f(x).
To solve f(x) = k, draw horizontal line y = k and read x-coordinates of intersections.
To solve f(x) = g(x), plot y = f(x) and y = g(x); solutions are x-coordinates of intersection points.
For equations not in the form 'graph = ...', rearrange to match the given graph. E.g., x² - 4x + 3 = 1 becomes x² - 4x - 2 = -4.
When solving for x, give only x-coordinates. Include y-coordinates only for simultaneous equations.

Solving f(x) = g(x) graphically

Example: Quadratic Graph from Table

Substitute each x-value into y = x² − 4x − 3 to complete the table of values: | x | −2 | −1 | 0 | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---|---|---|---| | y | 9 | 2 | −3 | −6 | −7 | −6 | −3 | 2 |
Plot the points and draw a smooth u-shaped curve for −2 ≤ x ≤ 5.
Solve x² − 4x − 3 = 0 by reading the x-intercepts from the graph: x ≈ −0.65 and x ≈ 4.65.
Line of symmetry: x = 2, the midpoint of the roots (and −b/(2a) = 4/2 = 2).

y = x² − 4x − 3

Example: Reciprocal Graph

Substitute each x-value into y = 15/x to complete the table of values: | x | −5 | −3 | −2 | −1 | 1 | 2 | 3 | 5 | |---|---|---|---|---|---|---|---|---| | y | −3 | −5 | −7.5 | −15 | 15 | 7.5 | 5 | 3 |
Draw two separate branches, one for x < 0 and one for x > 0; do not connect them at x = 0.
Solve 15/x = 6 by drawing the horizontal line y = 6 and reading the x-coordinate of the intersection: x = 2.5.
Reciprocal graphs have asymptotes at x = 0 and y = 0.

y = 15/x

Example: Solving with Horizontal Line

Substitute each x-value into y = 1 + 5x − x² to complete the table of values: | x | −1 | 0 | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---|---|---| | y | −5 | 1 | 5 | 7 | 7 | 5 | 1 |
Draw the horizontal line y = 3. The x-coordinates of the intersections solve 1 + 5x − x² = 3.
Solutions: approximately x = 0.44 and x = 4.56.
Line of symmetry: x = 2.5 (the midpoint of the two solutions above, and the x-coordinate of the vertex).

y = 1 + 5x − x²

Example: Solving with Another Line

Substitute each x-value into y = −x² + x + 5 to complete the table of values: | x | −3 | −2 | −1 | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---|---|---|---| | y | −7 | −1 | 3 | 5 | 5 | 3 | −1 | −7 |
Draw the line y = x. The x-coordinates of the intersections solve −x² + x + 5 = x, which simplifies to x² = 5.
Solutions: x = ±√5 ≈ ±2.24.
Vertex (turning point): (0.5, 5.25). Line of symmetry: x = 0.5.
Roots of −x² + x + 5 = 0: approximately x = −1.79 and x = 2.79.

y = −x² + x + 5

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साइन अप करें — फ्री लॉग इन

प्रैक्टिस सवाल

फ्री प्रीव्यू — 57 में से 8 सवाल। सभी देखने के लिए साइन अप करें।

1.Which of the following is the equation of a straight line?
Easy
- Ay = 3x + 2
- By = x² + 3x + 2
- Cy = 1/x
- Dy = -x² + 3x + 2
2.What is the name of the shape of a quadratic graph?
Easy
- AParabola
- BHyperbola
- CStraight line
- DCircle
3.For a quadratic graph y = ax² + bx + c, if a is positive, the graph is:
Easy
- Au-shaped
- Bn-shaped
- Ca straight line
- DL-shaped
4.What is the equation of the line of symmetry for a quadratic graph with vertex at (2, 5)?
Easy
- Ax = 2
- By = 2
- Cx = 5
- Dy = 5
5.Which of the following is a reciprocal graph?
Easy
- Ay = 4/x
- By = 4x + 1
- Cy = x² - 4
- Dy = -x² + 4
6.What is the value of y when x = -2 in the equation y = x² - 3x?
Easy
- A10
- B-2
- C-10
- D2
7.Which of the following is the equation of a horizontal line?
Easy
- Ay = 4
- Bx = 4
- Cy = x
- Dy = -x
8.The graph of y = -x² + 2x + 3 is:
Easy
- An-shaped
- Bu-shaped
- Ca straight line
- DL-shaped

और सवाल

9.A quadratic graph has a minimum point at (3, -4). What is the equation of its line of symmetry?
Easy
- Ax = 3
- By = -4
- Cx = -4
- Dy = 3
10.Using the graph of y = x² - 4x - 3, the solutions to x² - 4x - 3 = 0 are approximately:
Medium
- A-0.65 and 4.65
- B-1 and 5
- C0 and 4
- D-2 and 6
11.To solve x² - 4x - 2 = 5 using the graph of y = x² - 4x - 2, you should:
Medium
- ADraw the line y = 5 and find x-coordinates of intersection
- BDraw the line y = 0 and find x-coordinates of intersection
- CDraw the line x = 5 and find y-coordinates of intersection
- DDraw the line y = -5 and find x-coordinates of intersection
12.The graph of y = 10 - 8x² is symmetrical about:
Medium
- Athe y-axis
- Bthe x-axis
- Cthe line y = x
- Dthe line y = -x
13.The graph of y = 15/x for x > 0 is:
Medium
- Aa decreasing curve approaching both axes
- Ban increasing straight line
- Ca u-shaped curve
- Da horizontal line
14.Using the graph of y = 15/x, solve 15/x = 6. The solution is approximately:
Medium
- A2.5
- B0.4
- C90
- D21
15.The line y = x intersects the curve y = -x² + x + 5 at two points. Their x-coordinates satisfy:
Medium
- Ax² - 5 = 0
- Bx² + x + 5 = 0
- C-x² + 5 = 0
- Dx² + 2x + 5 = 0
16.What is the shape of the graph of y = x² + 3x - 4?
Easy
- AA straight line
- BA u-shaped parabola
- CAn n-shaped parabola
- DA reciprocal curve
17.Which of the following is the equation of a reciprocal graph?
Easy
- Ay = 3x + 2
- By = x² - 4
- Cy = 5/x
- Dy = -x + 1
18.The graph of y = x² - 4x + 1 has a vertical line of symmetry. What is its equation?
Medium
- Ax = 2
- Bx = -2
- Cx = 4
- Dx = -4
19.A quadratic graph has a maximum point at (1, 5). What is the equation of its line of symmetry?
Easy
- Ax = 5
- Bx = 1
- Cy = 5
- Dy = 1
20.Using the graph of y = x² - 2x - 3, the solutions to x² - 2x - 3 = 0 are approximately:
Medium
- A-1 and 3
- B1 and -3
- C2 and -1
- D-2 and 3
21.What is the y-intercept of the graph y = x² - 5x + 6?
Easy
- A6
- B-5
- C1
- D0
22.Which of the following graphs is a negative quadratic?
Easy
- Ay = x² + 2x + 1
- By = -x² + 3x - 2
- Cy = 2x - 5
- Dy = 4/x
23.The graph of y = 12/x for x > 0 is:
Medium
- AA straight line through the origin
- BA u-shaped curve
- CAn L-shaped curve in the first quadrant
- DAn n-shaped curve
24.Which of the following is the equation of a vertical line?
Easy
- Ay = 3
- Bx = -2
- Cy = x
- Dy = 2x + 1
25.The graph of y = -x² + 4x - 3 is:
Medium
- AA u-shaped parabola with minimum at (2,1)
- BAn n-shaped parabola with maximum at (2,1)
- CA u-shaped parabola with minimum at (2,-1)
- DAn n-shaped parabola with maximum at (2,-1)
26.A quadratic graph has equation y = x² - 4x + 3. What is the y-intercept of this graph?
Easy
- A3
- B-4
- C1
- D0
27.What is the shape of the graph of y = -x² + 3x - 2?
Easy
- AA straight line
- BA u-shape
- CAn n-shape
- DTwo L-shaped branches
28.A quadratic graph has a minimum point at (2, -5). What is the equation of its line of symmetry?
Easy
- Ax = -5
- Bx = 2
- Cy = -5
- Dy = 2
29.What is the equation of the vertical line passing through the point (3, -2)?
Easy
- Ax = 3
- By = -2
- Cy = 3
- Dx = -2
30.The graph of y = -x² + 2x + 3 has a maximum point. What are the coordinates of the maximum point?
Medium
- A(1, 4)
- B(-1, 0)
- C(1, 0)
- D(2, 3)
31.The graph of y = 12/x for x > 0 is drawn. Using the graph, solve 12/x = 8. The solution is approximately:
Medium
- Ax = 1.5
- Bx = 0.67
- Cx = 4
- Dx = 20
32.What is the shape of the graph of y = 2x² + 3x + 4?
Easy
- AA u-shape (positive parabola)
- BAn n-shape (negative parabola)
- CA straight line
- DTwo L-shaped branches
33.For a quadratic graph with equation y = -x² + 2x + 3, the line of symmetry is:
Medium
- Ax = 1
- Bx = -1
- Cx = 2
- Dx = 3
34.Which of the following are equations of reciprocal graphs? (Select all that apply)
Easy
- Ay = 5/x
- By = x² + 1
- Cy = 1/x
- Dy = 3x - 2
35.A positive quadratic graph has a maximum point.
Easy
True or false?
36.Match each equation to its graph type.
Easy
- y = 2x + 3
- y = x² - 1
- y = 4/x
- Straight line
- Quadratic
- Reciprocal
37.Arrange these steps in the correct order to solve x² - 4x - 2 = 5 using the graph of y = x² - 4x - 2.
Medium
- Draw the line y = 5 on the same axes
- Find the x-coordinates of intersection points
- Plot the graph of y = x² - 4x - 2
- Read off the x-values where the line meets the curve
38.Which of the following is the equation of a quadratic graph?
Easy
- Ay = 2x + 3
- By = x² - 5x + 6
- Cy = 4/x
- Dy = 7
39.A reciprocal graph has two curved branches that are L-shaped and never connect.
Easy
True or false?
40.The graph of y = -2x² + 4x - 1 is:
Easy
- Aa u-shape
- Ban n-shape
- Ca straight line
- Dtwo L-shaped branches
41.The graph of y = x² - 4x + 3 is drawn. What are the x-coordinates of the points where this graph intersects the line y = 3?
Medium
- A0 and 4
- B1 and 3
- C2 and 2
- D-1 and 5
42.Which of the following equations represents a reciprocal graph?
Easy
- Ay = 2x + 3
- By = x² - 1
- Cy = 5/x
- Dy = -x² + 2
43.Complete the table for y = x² - 2x. When x = -1, y = ?
Easy
44.Write down the equation of the vertical line of symmetry of the graph y = x² - 4x + 1.
Easy
45.Using the graph of y = 15/x, solve 15/x = 6. Give your answer to 2 decimal places.
Easy
46.Complete the sentence.
Easy
A quadratic graph always has a vertical line of ____.
47.Complete the table for y = x² - 2x.
Easy
When x = -1, y = ____.
48.Complete the table for y = 2x² - 3x. When x = -2, find y.
Easy
49.Fill in the blank.
Easy
The graph of y = x² - 6x + 8 has a line of symmetry at x = ____.
50.Using the graph of y = 12/x for x > 0, solve 12/x = 8. Give your answer to 2 decimal places.
Easy
51.Complete the sentence about the graph of y = 1/x.
Easy
The graph of y = 1/x has two curved branches that are ____ shaped and never connect.
52.A quadratic graph has equation y = x² - 4x + 1. What is the x-coordinate of its vertex?
Medium
53.Complete the table of values for y = x² - 3x.
Easy
When x = -1, y = ____.
54.Explain how to use the graph of y = x² - 4x - 2 to solve x² - 4x + 3 = 1.
Hard
55.Complete the table for y = x² - 2x. When x = -1, y = ?
Easy
- A2
- B5
- C4
- D3
56.Complete the table for y = 2x² - 3x. When x = -2, find y.
Easy
- A13
- B16
- C14
- D15
57.A quadratic graph has equation y = x² - 4x + 1. What is the x-coordinate of its vertex?
Medium
- A4
- B1
- C2
- D3

रिवीज़न नोट्स

## Further Graphs - Quick Revision ### Types of Graphs - **Straight line**: y = mx + c (e.g. y = 3x + 2). Also y = x, y = -x, horizontal y = c, vertical x = k. - **Quadratic**: y = ax² + bx + c. Shape: parabola. Positive a → u-shape (minimum). Negative a → n-shape (maximum). Key features: vertex (turning point), vertical line of symmetry x = k, roots (x-intercepts), y-intercept. - **Reciprocal**: y = a/x (x ≠ 0). Two L-shaped branches, never cross axes. For a > 0, branches in first and third quadrants. ### Drawing Graphs from Tables - Substitute x-values into equation to get y-values. Use brackets for negatives. - Plot points accurately (within half a square). Join with smooth freehand curve (no ruler). - For quadratics, expect symmetry. For reciprocals, omit x = 0. - Calculator tip: use table function (start, end, step) to generate values. ### Solving Equations from Graphs - To solve f(x) = 0: read x-intercepts (roots). - To solve f(x) = k: draw horizontal line y = k, read x-coordinates of intersections. - To solve f(x) = g(x): plot both graphs, read x-coordinates of intersection points. - For rearranged equations, rewrite as "graph = ..." and proceed as above. ### Key Formulas - Quadratic: y = ax² + bx + c - Reciprocal: y = a/x - Line of symmetry: x = k (k is x-coordinate of vertex) ### Exam Tips - Check plotted points match given table values. - If a point doesn't fit the shape, recalculate. - When solving, give only x-values unless simultaneous equations require both coordinates.

Quadratic graph: A graph of the form y = ax² + bx + c, producing a parabola (u-shape or n-shape).
Reciprocal graph: A graph of the form y = a/x (x ≠ 0), consisting of two L-shaped branches.
Vertex: The turning point of a quadratic graph; minimum for positive quadratics, maximum for negative quadratics.
Line of symmetry: A vertical line x = k that divides a quadratic graph into two mirror-image halves; k is the x-coordinate of the vertex.
Roots: The x-intercepts of a graph, where y = 0; solutions to the equation f(x) = 0.
y-intercept: The point where the graph crosses the y-axis (x = 0).
Table of values: A set of x and y coordinates generated by substituting x-values into an equation, used to plot the graph.
Point of intersection: A point where two graphs meet; its coordinates satisfy both equations simultaneously.

फ्लैशकार्ड

What is the general equation of a straight line?

y = mx + c

What is the equation of a horizontal line?

y = c

What is the equation of a vertical line?

x = k

What is the general equation of a quadratic graph?

y = ax² + bx + c

What shape does a positive quadratic graph have?

A u-shape (minimum point at the bottom).

What shape does a negative quadratic graph have?

An n-shape (maximum point at the top).

What is the vertex of a quadratic graph?

The point where the graph turns; the minimum or maximum point.

What is the line of symmetry of a quadratic graph?

A vertical line through the vertex, equation x = k, where k is the x-coordinate of the vertex.

What are the roots of a quadratic graph?

The x-intercepts where the graph crosses the x-axis.

What is the general equation of a reciprocal graph?

y = a/x, where x ≠ 0.

What is the shape of a reciprocal graph y = a/x for a > 0?

Two L-shaped curved branches, one in the first quadrant and one in the third quadrant.

Why can't x = 0 be used in a reciprocal graph?

Division by zero is undefined, so there is no value at x = 0.

How do you draw a graph from a table of values?

Plot the (x, y) points accurately, then join them with a single smooth freehand curve.

What should you be careful about when substituting negative numbers into a quadratic equation?

Always put brackets around negative numbers and use BIDMAS.

How can you solve x² - 4x - 3 = 0 using its graph?

Read the x-coordinates where the graph crosses the x-axis (the roots).

How can you solve x² - 4x - 2 = 5 using the graph of y = x² - 4x - 2?

Draw the horizontal line y = 5 and read the x-coordinates of the intersection points.

How can you solve x² - 4x - 2 = x + 1 using the graph of y = x² - 4x - 2?

Plot the line y = x + 1 on the same axes and read the x-coordinates of the intersection points.

How do you solve a different equation like x² - 4x + 3 = 1 using the graph of y = x² - 4x - 2?

Rearrange to get 'graph = ...' by subtracting 5 from both sides: x² - 4x - 2 = -4, then use the graph and line y = -4.

What are the coordinates of intersection of two graphs?

The points where the two graphs meet; the x and y coordinates satisfy both equations.

How do you solve simultaneous equations graphically?

Plot both equations on the same axes, find their point(s) of intersection, and read the x and y coordinates.

What is the equation of the line of symmetry for y = x² - 3x - 6?

x = 1.5 (the x-coordinate of the vertex).

What is the equation of the line of symmetry for y = 7 + 2x - x²?

x = 1 (the x-coordinate of the vertex).

What is the highest point of a negative quadratic graph called?

The maximum point or vertex.

What is the lowest point of a positive quadratic graph called?

The minimum point or vertex.

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