NOTES:
Given a quadratic function in standard format (y = ax² + bx + c), the direction of the parabola is as follows:
- if "a" is positive, then opens UP
- if "a" is negative, then opens DOWN
Given a quadratic function in standard format (y = ax² + bx + c), the vertex can be found as follows:
- the Axis Of Symmetry (x-value) is: x = [tex]\frac{-b}{2a}[/tex]
- y-value is found by plugging in the AOS for "x" in the equation
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1) y = x² + 11x + 24
- a = +1 so the parabola opens UP
- x = [tex]\frac{-b}{2a}[/tex] = [tex]\frac{-11}{2(1)}[/tex] = [tex]-\frac{11}{2}[/tex]
- y = [tex](-\frac{11}{2})^{2}[/tex] + 11[tex](-\frac{11}{2} )[/tex] + 24 = [tex]-\frac{25}{4}[/tex]
- vertex [tex](-\frac{11}{2}[/tex], [tex]-\frac{25}{4})[/tex] is in Quadrant 3 and is below the x-axis
This COULD be the graph of the rain gauge.
The graph should contain the vertex, x-intercepts (-3, 0) and (-8, 0), and y-intercept (0, 24)
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2) y = -x² - 6x - 8
- a = -1 so the parabola opens DOWN
- x = [tex]\frac{-b}{2a}[/tex] = [tex]\frac{-(-6)}{2(-1)}[/tex] = [tex]\frac{6}{-2}[/tex] = -3
- y = -(-3)² - 6(-3) - 8 = -9 + 18 - 8 = 1
- vertex (-3, 1) is in Quadrant 2 and is above the x-axis
This could NOT be the graph of the rain gauge.
The graph should contain the vertex, x-intercepts (-2, 0) and (-4, 0), and y-intercept (0, -8)
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3) y = x² - 2x + 3
- a = 1 so the parabola opens UP
- x = [tex]\frac{-b}{2a}[/tex] = [tex]\frac{-(-2)}{2(1)}[/tex] = [tex]\frac{2}{2}[/tex] = 1
- y = (1)² - 2(1) + 3 = 1 - 2 + 3 = 2
- vertex (1, 2) is in Quadrant 1 and is above the x-axis
This could NOT be the graph of the rain gauge.
The graph should contain the vertex, y-intercept (0, 3), and its mirror image (2, 3). There are no x-intercepts
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4) y = x² + 4x + 4
- a = 1 so the parabola opens UP
- x = [tex]\frac{-b}{2a}[/tex] = [tex]\frac{-4}{2(1)}[/tex] = -2
- y = (-2)² + 4(-2) + 4 = 4 - 8 + 4 = 0
- vertex (-2, 0) is in Quadrant 2 and is on the x-axis
This COULD be the graph of the rain gauge.
The graph should contain the vertex, y-intercept (0, 4), and its mirror image (-4, 4). The x-intercept is the vertex.
Compared to the other four graphs, this is most likely the equation for the rain gauge!
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5) y = 3x² + 21x + 30
- a = +3 so the parabola opens UP
- x = [tex]\frac{-b}{2a}[/tex] = [tex]\frac{-21}{2(3)}[/tex] = [tex]-\frac{7}{2}[/tex]
- y = 3[tex](-\frac{7}{2})^{2}[/tex] + 21[tex](-\frac{7}{2} )[/tex] + 30 = [tex]-\frac{27}{4}[/tex]
- vertex [tex](-\frac{7}{2}[/tex], [tex]-\frac{27}{4})[/tex] is in Quadrant 3 and is below the x-axis
This COULD be the graph of the rain gauge.
The graph should contain the vertex, x-intercepts (-2, 0) and (-5, 0), and y-intercept (0, 30)
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