Quadratic graphs and their properties

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レッスンノート

The shape of a quadratic graph

A quadratic has the form y = ax² + bx + c with a \neq 0. Its graph is a smooth, symmetrical curve called a parabola.
If a > 0 the parabola is U-shaped with a minimum; if a < 0 it is ∩-shaped with a maximum.
The y-intercept is c, where x = 0.

Positive and negative leading coefficients

Roots and the line of symmetry

Real roots are the x-values where the graph meets the x-axis, so y = 0. It may cross the axis or touch it at a repeated root.
The parabola is symmetrical about a vertical line of symmetry halfway between the roots: x = \frac{x₁ + x₂}{2}, or x = -\frac{b}{2a}.
A quadratic may have two roots, one repeated root (the graph just touches the axis), or no real roots (the graph misses the axis).

Roots and turning point

The turning point

The turning point (vertex) lies on the line of symmetry. Substitute that x-value into the equation to find y.
For y = x² - 4x + 3: roots at x = 1 and x = 3, line of symmetry x = 2, turning point y = 4 - 8 + 3 = -1, so (2, -1).
Completed square form: y = (x - p)² + q has turning point (p, q).

Drawing a quadratic

Make a table of values for several x values (include negatives), plot the points and join them with a smooth curve, not straight lines.
Check the curve is symmetrical and passes through the intercepts.

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練習問題

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1.What is the graph of a quadratic function called?
Easy
- AA hyperbola
- BA parabola
- CA circle
- DA straight line
2.For y = x² - 5, where does the graph cross the y-axis?
Easy
- A(0, 5)
- B(0, -5)
- C(5, 0)
- D(-5, 0)
3.What shape is y = 2x² + 1?
Easy
- A∩-shaped
- BU-shaped
- Ca straight line
- Dan S-shape
4.What shape is y = 3 - x²?
Easy
- AU-shaped
- B∩-shaped
- Ca straight line
- Da circle
5.What are the roots of y = x² - 9?
Easy
- Ax = 3 only
- Bx = \pm 3
- Cx = \pm 9
- Dx = 0
6.The roots of a quadratic are x = 1 and x = 5. What is the line of symmetry?
Easy
- Ax = 2
- Bx = 3
- Cx = 4
- Dx = 6
7.The roots of a quadratic are x = -2 and x = 6. What is the line of symmetry?
Easy
- Ax = 1
- Bx = 2
- Cx = 4
- Dx = 3
8.What is the line of symmetry of y = x² + 6x + 1?
Easy
- Ax = 3
- Bx = -3
- Cx = 6
- Dx = -6

もっと問題

9.What is y when x = 3 on y = x² - 2x?
Easy
- A3
- B0
- C5
- D9
10.What is the turning point of y = x² + 2?
Easy
- A(2, 0)
- B(0, 2)
- C(0, -2)
- D(1, 3)
11.What is the turning point of y = x² - 4x + 3?
Medium
- A(2, -1)
- B(2, 1)
- C(-2, -1)
- D(4, 3)
12.What is the turning point of y = (x + 1)² - 4?
Medium
- A(1, -4)
- B(-1, -4)
- C(-1, 4)
- D(1, 4)
13.The turning point of y = x² - 6x + 5 is:
Medium
- A(3, -4)
- B(3, 4)
- C(-3, -4)
- D(6, 5)
14.How many real roots does y = x² + 1 have?
Medium
- A2
- B1
- C0
- Dinfinitely many
15.How many roots does a parabola that just touches the x-axis have?
Medium
- A0
- B1 (repeated)
- C2
- D3
16.A parabola has its minimum at (2, -3). How many roots does it have?
Medium
- A0
- B1
- C2
- D3
17.Which equation could match a graph with roots at x = 2 and x = 5 that is U-shaped?
Medium
- Ay = (x - 2)(x - 5)
- By = (x + 2)(x + 5)
- Cy = -(x - 2)(x - 5)
- Dy = x² + 10
18.Solve x² - 5x + 6 = 0 to find where y = x² - 5x + 6 crosses the x-axis.
Medium
- Ax = 2, 3
- Bx = -2, -3
- Cx = 1, 6
- Dx = 5, 6
19.What is the minimum value of y = x² - 6x + 11?
Medium
- A2
- B11
- C-2
- D3
20.What is the maximum value of y = 5 - (x - 1)²?
Medium
- A1
- B5
- C-1
- D6
21.The graph of y = ax² + bx + c crosses the y-axis at (0, 7). What is c?
Medium
- A0
- B7
- C-7
- D\frac{1}{7}
22.When plotting a quadratic from a table of values, you should join the points with:
Easy
- Aruled straight lines
- Ba smooth curve
- Ca zigzag
- Ddotted lines only
23.Which x value is NOT needed in a table for y = x² - 2x, from -2 to 4?
Easy
- A-2
- B1
- C3
- D10
24.Which of these quadratics has y always positive?
Medium
- Ay = x² - 4
- By = x² + 4
- Cy = x² - 4x
- Dy = -x² + 4
25.y = x² is symmetrical about which line?
Easy
- Ay = x
- Bthe y-axis
- Cthe x-axis
- Dx = 1
26.A quadratic graph is a straight line.
Easy
True or false?
27.If the coefficient of x² is negative the graph has a maximum point.
Easy
True or false?
28.The real roots of a quadratic are the x-values where its graph meets the x-axis, either crossing or touching it.
Easy
True or false?
29.A quadratic always has two real roots.
Easy
True or false?
30.The turning point lies on the line of symmetry.
Easy
True or false?
31.y = (x - 2)² + 3 has its turning point at (-2, 3).
Medium
True or false?
32.Which statements about y = x² - 4x + 3 are true?
Medium
- AIt crosses the x-axis at 1
- BIt crosses the x-axis at 3
- CIts turning point is a maximum
- DIt crosses the y-axis at (0, 3)
- EIts line of symmetry is x = 3
33.Which graphs are ∩-shaped?
Medium
- Ay = -x²
- By = x² - 1
- Cy = 4 - x²
- Dy = 2x²
- Ey = -3x² + x
34.Find the turning point of y = x² - 6x + 5 and say whether it is a maximum or minimum.
Medium

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