Further Graphs

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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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Câu hỏi luyện tập

Xem trước miễn phí — 8 trên 57 câu hỏi. Đăng ký để xem tất cả.

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

Thêm câu hỏi

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

Ghi chú ôn tập

## 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.

Thẻ ghi nhớ

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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