Quadratic Graphs

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Key Features of Quadratic Graphs

A quadratic graph has equation y = ax² + bx + c with a ≠ 0.
It is a smooth curve called a parabola, with a vertical line of symmetry.
If a > 0, the graph is u-shaped (minimum turning point).
If a < 0, the graph is n-shaped (maximum turning point).
The y-intercept is at (0, c).
The x-intercepts (roots) are solutions to ax² + bx + c = 0; there can be 0, 1, or 2 roots.
The turning point (vertex) is the minimum or maximum point.

Positive and negative quadratics

Sketching a Quadratic Graph

Draw axes and mark the y-intercept (0, c).
Find and mark the roots by solving ax² + bx + c = 0 (factorising, completing square, or quadratic formula).
Determine the shape: u-shaped if a > 0, n-shaped if a < 0.
Sketch a smooth curve through the intercepts, showing the turning point if known.
Label all intercepts and the turning point coordinates.

Key features of a quadratic graph

Finding the Turning Point by Completing the Square

Rewrite y = ax² + bx + c as y = a(x - p)² + q.
The turning point is at (p, q) (note sign change for p).
For y = (x - 3)² + 2, the minimum is (3, 2).
For y = (x + 3)² + 2, the minimum is (-3, 2).
The value of a does not affect the turning point coordinates but affects the shape.

Finding the Turning Point by Differentiation

Differentiate y = ax² + bx + c to get dy/dx = 2ax + b.
Set dy/dx = 0 and solve for x to find the x-coordinate of the turning point.
Substitute this x into the original equation to find the y-coordinate.
This method works for any quadratic and confirms whether it is a maximum or minimum.

Finding the Equation of a Quadratic from Its Graph

If the vertex (p, q) and one other point are known, use y = a(x - p)² + q.
Substitute the other point to find a.
If the roots (x₁, 0) and (x₂, 0) and one other point are known, use y = a(x - x₁)(x - x₂).
Substitute the other point to find a.
If a = 1, only the vertex or roots are needed.

Example: Sketching y = x² - 5x + 6

y-intercept: (0, 6) (c = 6).
Factorise: y = (x - 2)(x - 3) → roots at (2, 0) and (3, 0).
a = 1 > 0, so graph is u-shaped.
Sketch a smooth u-shaped curve through (0,6), (2,0), (3,0).

y = x² − 5x + 6

Example: Sketching y = x² - 6x + 13

y-intercept: (0, 13).
Complete square: y = (x - 3)² + 4 → vertex at (3, 4) (minimum).
Since vertex is above x-axis and a > 0, there are no real roots.
Sketch u-shaped curve with vertex (3,4) and y-intercept (0,13).

y = x² − 6x + 13

Example: Sketching y = -x² - 4x - 4

y-intercept: (0, -4).
Differentiate: dy/dx = -2x - 4; set to 0 → x = -2.
Substitute: y = -(-2)² - 4(-2) - 4 = 0 → vertex at (-2, 0) (maximum).
Only one root at x = -2 (touches x-axis).
Sketch n-shaped curve with vertex (-2,0) and y-intercept (0,-4).

y = −x² − 4x − 4

Example: Finding Equation from Roots

Given roots at x = 2 and x = 3, and point (0, 24).
Use y = a(x - 2)(x - 3); substitute (0,24): 24 = a(-2)(-3) = 6a → a = 4.
Equation: y = 4(x - 2)(x - 3) or y = 4x² - 20x + 24.

y = 4(x − 2)(x − 3)

Example: Finding Equation from Vertex

Given vertex at (9, -16) and point (2, 82).
Use y = a(x - 9)² - 16; substitute (2,82): 82 = a(2-9)² - 16 = 49a - 16 → 49a = 98 → a = 2.
Equation: y = 2(x - 9)² - 16 or y = 2x² - 36x + 146.

y = 2(x − 9)² − 16

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1.What is the shape of the graph of y = x² + 3x - 4?
Easy
- AU-shaped
- BN-shaped
- CStraight line
- DS-shaped
2.The y-intercept of y = x² - 5x + 6 is at (0, c). What is c?
Easy
- A6
- B-5
- C0
- D-6
3.How many x-intercepts can a quadratic graph have?
Easy
- A0, 1 or 2
- BAlways 2
- CAlways 1
- D0 or 2
4.The turning point of y = (x - 3)² + 5 has coordinates:
Easy
- A(3, 5)
- B(-3, 5)
- C(3, -5)
- D(-3, -5)
5.Complete the square: x² + 6x + 11 = (x + a)² + b. Find a and b.
Medium
- Aa = 3, b = 2
- Ba = -3, b = 2
- Ca = 3, b = 20
- Da = -3, b = 20
6.The graph of y = -x² + 4x - 3 has a maximum point. What is its x-coordinate?
Medium
- A2
- B-2
- C4
- D-4
7.A quadratic graph has roots at x = 1 and x = 5. Which equation could represent it?
Easy
- Ay = (x - 1)(x - 5)
- By = (x + 1)(x + 5)
- Cy = (x - 1)(x + 5)
- Dy = (x + 1)(x - 5)
8.The curve y = x² - 6x + 13 has a turning point at (3, 4). How many x-intercepts does it have?
Medium
- A0
- B1
- C2
- DCannot be determined

Thêm câu hỏi

9.Find the equation of the quadratic with vertex (2, -3) and passing through (0, 5).
Hard
- Ay = 2(x - 2)² - 3
- By = 2(x + 2)² - 3
- Cy = -2(x - 2)² - 3
- Dy = (x - 2)² - 3
10.The graph of y = 24 + 5x - x² has roots at x = -3 and x = 8. What is the y-intercept?
Easy
- A24
- B-24
- C5
- D-5
11.What is the name of the shape of a quadratic graph?
Easy
- AParabola
- BHyperbola
- CEllipse
- DCircle
12.For a quadratic graph y = ax² + bx + c, if a < 0, the graph is:
Easy
- AN-shaped
- BU-shaped
- CStraight line
- DHorizontal
13.The axis of symmetry of y = (x + 2)² - 3 is:
Easy
- Ax = -2
- Bx = 2
- Cx = -3
- Dx = 3
14.Which of the following is the turning point of y = x² + 4x - 3?
Medium
- A(-2, -7)
- B(2, 9)
- C(-2, -11)
- D(2, 5)
15.The curve y = 2(x - 1)² - 8 has x-intercepts at:
Medium
- Ax = -1 and x = 3
- Bx = 1 and x = -3
- Cx = -1 and x = -3
- Dx = 1 and x = 3
16.A quadratic graph passes through (0, -6), (2, 0) and (4, 0). Which equation represents it?
Hard
- Ay = -0.75(x - 2)(x - 4)
- By = 0.75(x - 2)(x - 4)
- Cy = -0.75(x + 2)(x + 4)
- Dy = 0.75(x + 2)(x + 4)
17.Write x² + 10x + 14 in the form (x + a)² + b. What are the values of a and b?
Medium
- Aa = 5, b = -11
- Ba = 5, b = 39
- Ca = 10, b = 14
- Da = -5, b = -11
18.Find the turning point of y = x² + 4x - 3 by completing the square.
Medium
- A(-2, -7)
- B(2, 9)
- C(-2, -11)
- D(2, 5)
19.The diagram shows a sketch of the curve y = x² + 3x - 4. Find the coordinates of the points A, B and C (A and B are x-intercepts, C is y-intercept). Which option gives the correct coordinates?
Medium
- AA(-4,0), B(1,0), C(0,-4)
- BA(-1,0), B(4,0), C(0,-4)
- CA(-4,0), B(1,0), C(0,4)
- DA(-1,0), B(4,0), C(0,4)
20.Factorise 24 + 5x - x².
Medium
- A(8 - x)(3 + x)
- B(8 + x)(3 - x)
- C(6 - x)(4 + x)
- D(6 + x)(4 - x)
21.The diagram shows a sketch of y = 24 + 5x - x². The curve crosses the x-axis at a and b (a < b) and the y-axis at c. Work out the values of a, b and c.
Medium
- Aa = -3, b = 8, c = 24
- Ba = -8, b = 3, c = 24
- Ca = -3, b = 8, c = -24
- Da = -8, b = 3, c = -24
22.Sketch the graph of y = (x - 1)². What is the turning point?
Easy
- A(1, 0)
- B(-1, 0)
- C(0, 1)
- D(0, -1)
23.Write 6x - 0.5x² - 7 in the form y = a(x - p)² + q.
Hard
- Ay = -0.5(x - 6)² + 11
- By = -0.5(x - 6)² - 7
- Cy = -0.5(x - 6)² + 25
- Dy = -0.5(x - 6)² - 25
24.For the curve y = 6x - 0.5x² - 7, what are the coordinates of the turning point?
Medium
- A(6, 11)
- B(6, -7)
- C(-6, 11)
- D(-6, -7)
25.State the nature of the turning point of y = 6x - 0.5x² - 7.
Easy
- AMaximum
- BMinimum
- CNeither
- DCannot be determined
26.State the equation of the axis of symmetry of y = 6x - 0.5x² - 7.
Medium
- Ax = 6
- Bx = -6
- Cy = 11
- Dy = -7
27.The curve C has equation y = 4(x - 1)² - a where a > 4. Using the axes below, sketch the curve C. On your sketch show clearly, in terms of a, the coordinates of the turning point. What are the coordinates?
Easy
- A(1, -a)
- B(1, a)
- C(-1, -a)
- D(-1, a)
28.The curve C has equation y = x² - 6x + 4. Find the coordinates of the turning point.
Medium
- A(3, -5)
- B(-3, 31)
- C(3, 13)
- D(-3, -5)
29.What is the y-intercept of the graph y = x² - 5x + 6?
Easy
- A(0, 6)
- B(0, -5)
- C(0, 5)
- D(0, -6)
30.Sketch the graph of y = x² - 5x + 6. What are the x-intercepts?
Easy
- A(2, 0) and (3, 0)
- B(-2, 0) and (-3, 0)
- C(1, 0) and (6, 0)
- D(-1, 0) and (-6, 0)
31.Find the equation of the graph that has roots at x = 2 and x = 3 and passes through (0, 24).
Medium
- Ay = 4(x - 2)(x - 3)
- By = 2(x - 2)(x - 3)
- Cy = 4(x + 2)(x + 3)
- Dy = 2(x + 2)(x + 3)
32.Find the equation of the graph with vertex (9, -16) that passes through (2, 82).
Medium
- Ay = 2(x - 9)² - 16
- By = 2(x + 9)² - 16
- Cy = 4(x - 9)² - 16
- Dy = 4(x + 9)² - 16
33.Write 6x - 0.5x² - 7 in the form a(x - p)² + q. What are the values of a, p and q?
Medium
- Aa = -0.5, p = 6, q = 11
- Ba = -0.5, p = 6, q = -7
- Ca = -0.5, p = -6, q = 11
- Da = 0.5, p = 6, q = 11
34.The curve y = 6x - 0.5x² - 7 has a turning point. What are its coordinates?
Medium
- A(6, 11)
- B(6, -7)
- C(-6, 11)
- D(6, 18)
35.The curve C has equation y = 4(x - 1)² - a where a > 4. Which of the following statements about the graph is correct?
Medium
- AThe turning point is at (1, -a) and the y-intercept is at (0, 4 - a)
- BThe turning point is at (-1, -a) and the y-intercept is at (0, 4 - a)
- CThe turning point is at (1, -a) and the y-intercept is at (0, 4 + a)
- DThe turning point is at (1, a) and the y-intercept is at (0, 4 - a)
36.A quadratic graph has vertex at (9, -16) and passes through (2, 82). What is its equation?
Medium
- Ay = 2(x - 9)² - 16
- By = 2(x + 9)² - 16
- Cy = (x - 9)² - 16
- Dy = 2(x - 9)² + 16
37.What is the shape of the graph of a quadratic equation called?
Easy
- AParabola
- BHyperbola
- CCircle
- DEllipse
38.The y-intercept of the graph y = x² - 5x + 6 is at (0, c). What is c?
Easy
- A6
- B-5
- C1
- D0
39.What is the y-intercept of the graph y = x² + 3x - 4?
Easy
- A(0, -4)
- B(0, 4)
- C(0, 3)
- D(0, -3)
40.A quadratic graph has equation y = x² + 2x - 3. What is its y-intercept?
Easy
- A(0, -3)
- B(0, 2)
- C(0, 3)
- D(0, 1)
41.Match each quadratic graph form to its turning point.
Easy
- y = (x-2)² + 3
- y = (x+1)² - 4
- y = -(x-5)² + 2
- (2, 3)
- (-1, -4)
- (5, 2)
42.The graph of y = -x² + 4x - 7 has a minimum point.
Medium
True or false?
43.Arrange the steps to sketch the graph of y = x² - 4x + 3 in the correct order.
Medium
- Find the y-intercept (0, 3).
- Find the roots by solving x² - 4x + 3 = 0.
- Determine that a > 0 so the graph is u-shaped.
- Sketch a smooth curve through the intercepts.
44.A quadratic graph has vertex at (-3, 5) and passes through (0, -4). What is its equation?
Hard
- Ay = (x+3)² + 5
- By = -(x+3)² + 5
- Cy = (x-3)² + 5
- Dy = -(x-3)² + 5
45.The graph of y = 2x² - 4x + 5 has how many x-intercepts?
Medium
- A0
- B1
- C2
- D3
46.Arrange these quadratic equations in order of number of x-intercepts (from least to greatest).
Medium
- y = x² + 4x + 5
- y = x² - 6x + 9
- y = x² - 4
47.The graph of y = x² - 4x + 3 crosses the y-axis at (0, c). What is c?
Easy
- A3
- B-3
- C1
- D4
48.A quadratic graph has equation y = -x² + 6x - 8. What is the nature of its turning point?
Medium
- AMaximum
- BMinimum
- CNeither
- DBoth
49.Which of the following are true for the graph of y = 3(x - 1)² + 2? (Select all that apply)
Hard
- AThe turning point is at (1, 2)
- BThe graph has a minimum point
- CThe y-intercept is 5
- DThe graph crosses the x-axis
50.Find the y-intercept of the graph y = x² - 5x + 6.
Easy
51.Complete the square: x² + 6x + 11 = (x + a)² + b.
Medium
a = ____ and b = ____
52.Find the x-coordinate of the turning point of y = 3x² - 12x + 7 using differentiation.
Medium
53.Complete the sentence.
Easy
The shape of a quadratic graph is called a ____.
54.Write down the equation of the axis of symmetry of y = (x+4)² - 9.
Easy
55.Find the x-coordinate of the turning point of y = 2x² - 8x + 5 by completing the square.
Medium
56.Write down the equation of the axis of symmetry of the graph y = (x + 3)² - 5.
Easy
57.Find the y-intercept of the graph y = x² - 5x + 6.
Easy
- A8
- B7
- C5
- D6
58.Find the x-coordinate of the turning point of y = 3x² - 12x + 7 using differentiation.
Medium
- A3
- B2
- C4
- D1
59.Find the x-coordinate of the turning point of y = 2x² - 8x + 5 by completing the square.
Medium
- A4
- B2
- C3
- D1

Ghi chú ôn tập

## Quadratic Graphs - Night Before Exam Summary ### Key Features - **General form**: \(y = ax^2 + bx + c\), \(a \neq 0\) - Shape: \(a > 0\) → **u-shaped** (minimum turning point); \(a < 0\) → **n-shaped** (maximum turning point) - The curve is a **parabola** with a vertical line of symmetry. - **y-intercept**: \((0, c)\) - **Roots** (x-intercepts): solutions to \(ax^2 + bx + c = 0\) (0, 1, or 2 real roots) ### Completing the Square - Write \(y = a(x - p)^2 + q\) - **Turning point** (vertex): \((p, q)\) (sign change for x!) - Example: \(y = (x-3)^2 + 2\) → minimum at \((3,2)\) - \(y = (x+3)^2 + 2\) → minimum at \((-3,2)\) - \(a\) does not affect vertex coordinates, only shape. ### Differentiation (alternative for turning point) - \(\frac{dy}{dx} = 2ax + b\) - Set \(\frac{dy}{dx} = 0\) → solve for \(x\) - Substitute back to find \(y\). ### Sketching Steps 1. Draw axes. 2. Mark y-intercept \((0,c)\). 3. Find and mark roots (if any). 4. Determine shape (u or n). 5. Find turning point (by completing square or differentiation). 6. Draw smooth curve through intercepts and vertex. ### Finding Equation from Graph - **If vertex \((p,q)\) and another point known**: use \(y = a(x-p)^2 + q\), substitute to find \(a\). - **If roots \(x_1, x_2\) and another point known**: use \(y = a(x-x_1)(x-x_2)\), substitute to find \(a\). ### Common Exam Tips - Always label intercepts and turning point coordinates. - For \(y = x^2 + 10x + 14\), complete square: \((x+5)^2 - 11\), vertex \((-5,-11)\). - For \(y = 6x - 0.5x^2 - 7\), rewrite as \(-0.5(x-6)^2 + 11\), vertex \((6,11)\), maximum. - Axis of symmetry: \(x = p\) from vertex form.

Quadratic graph: A graph of a quadratic function \(y = ax^2 + bx + c\) where \(a \neq 0\). It is a smooth curve called a parabola.
Parabola: The U-shaped or n-shaped curve of a quadratic graph.
Turning point (vertex): The minimum point (if \(a > 0\)) or maximum point (if \(a < 0\)) of a quadratic graph. Coordinates can be found by completing the square: \((p, q)\) from \(y = a(x-p)^2 + q\).
Roots: The x-values where the graph crosses the x-axis; solutions to \(ax^2 + bx + c = 0\).
y-intercept: The point where the graph crosses the y-axis; \((0, c)\).
Completing the square: Rewriting \(ax^2 + bx + c\) as \(a(x-p)^2 + q\) to find the turning point.
Axis of symmetry: The vertical line through the turning point, given by \(x = p\) in vertex form.

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