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Quadratic graphs and their roots
The graph of a quadratic is a smooth U-shaped curve called a parabola. Its key points tell you a lot.
The shape
y = x2 + bx + c makes a U shape. If the x2 term is negative, the curve is upside down (an ∩ shape).
It is symmetrical about a vertical line through its turning point.
It is symmetrical about a vertical line through its turning point.
Roots and intercept
The roots are where the curve crosses the x-axis (y = 0). Solve x2 + bx + c = 0.
The y-intercept is c (put x = 0).
The turning point is halfway between the roots.
The y-intercept is c (put x = 0).
The turning point is halfway between the roots.
Example
y = x2 − 2x − 8 = (x − 4)(x + 2)
Roots: x = 4 and x = −2. y-intercept: −8.
Line of symmetry: x = 1 (halfway between −2 and 4). Turning point: (1, −9).
Roots: x = 4 and x = −2. y-intercept: −8.
Line of symmetry: x = 1 (halfway between −2 and 4). Turning point: (1, −9).
Find the turning point of y = x2 − 6x + 5.
- Roots: (x − 1)(x − 5) = 0, so x = 1 and x = 5.
- Halfway: x = 3.
- y = 9 − 18 + 5 = −4
Answer: (3, −4)
A quadratic graph is a parabola. Roots are where y = 0, the y-intercept is c, and the turning point lies on the line of symmetry halfway between the roots.
The interactive lesson includes the diagrams for this topic.
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