constructing parabolas from quadratics you need the following items to construct a parabola line of...

Constructing Parabolas from QuadraticsConstructing Parabolas from Quadratics

You need the following items to construct You need the following items to construct a parabolaa parabola

• Line of Symmetry (axis of symmetry)Line of Symmetry (axis of symmetry)• VertexVertex• Direction of openingDirection of opening• Min/max vertex valueMin/max vertex value• Y-interceptY-intercept• X-interceptsX-intercepts

Plot all these points and drawPlot all these points and draw


When you identify you a, b, & c, plug in When you identify you a, b, & c, plug in the values to each partthe values to each part

• Line of Symmetry (axis of symmetry)Line of Symmetry (axis of symmetry)

x = x = -b-b

2a2a


•VertexVertex Plug in your value of Plug in your value of xx from from the line of symmetry and the line of symmetry and find your value of find your value of yy


•Direction of openingDirection of opening If If aa > 0 > 0

Then the parabola opens Then the parabola opens upup

If If aa < 0 < 0

Then the parabola opens Then the parabola opens downdown


•Minimum/Maximum vertex valueMinimum/Maximum vertex value

If If aa > 0 > 0

Then the vertex is the Then the vertex is the minimumminimum value value

If If aa < 0 < 0

Then the vertex is the Then the vertex is the maximummaximum value value


•Y-interceptY-intercept

To find the y-intercept,To find the y-intercept,

Set Set xx = 0 and = 0 and

solve of solve of yy


•x-interceptsx-intercepts To find the x-intercepts To find the x-intercepts

(don’t forget there can be 2 x-(don’t forget there can be 2 x-intercepts)intercepts)

Set y = 0 and Set y = 0 and

solve of x solve of x


Remember the 3 ways to Remember the 3 ways to find the x-intercepts find the x-intercepts

• FactoringFactoring• Completing the squareCompleting the square•Using the quadratic Using the quadratic formula formula


Remember to determine if there are REAL values of Remember to determine if there are REAL values of x, you use the discriminant x, you use the discriminant

√ √ b-4ac b-4ac If If discriminant discriminant > 0, > 0,

there are there are 2 REAL values 2 REAL values ofof x x

If If discriminant discriminant = 0, = 0,

therethere is 1 REAL value is 1 REAL value ofof x x

If If discriminant discriminant < 0,< 0,

there are there are NO REAL valuesNO REAL valuesofof x x

Constructing Parabolas from Constructing Parabolas from QuadraticsQuadratics

Example: Example:

Construct the parabola for the Construct the parabola for the following quadratic equationfollowing quadratic equation

y = x + x - 5y = x + x - 5


y = x + x – 5y = x + x – 5

a = 1, b= 1 c= -5a = 1, b= 1 c= -5

Line of symmetryLine of symmetry

x = x = -1-1 x = -1/2 x = -1/2

2(1)2(1)


y = x + x – 5y = x + x – 5

a = 1, b= 1 c= -5a = 1, b= 1 c= -5

Vertex Vertex

y = (1/2) - ½ - 5y = (1/2) - ½ - 5

y = ¼ - ½ - 5y = ¼ - ½ - 5

y = ¼ - 2/4 – 20/4y = ¼ - 2/4 – 20/4

y = -21/4y = -21/4

Therefore, the vertex is (-1/2, -21/4)Therefore, the vertex is (-1/2, -21/4)


y = x + x – 5y = x + x – 5

a = 1, b= 1 c= -5a = 1, b= 1 c= -5

Direction of openingDirection of opening

a > 0a > 0

Therefore, the parabola opens upTherefore, the parabola opens up


y = x + x – 5y = x + x – 5

a = 1, b= 1 c= -5a = 1, b= 1 c= -5

Min/Mix Vertex ValueMin/Mix Vertex Value

a > 0a > 0

Therefore, the vertex value is the Therefore, the vertex value is the minimum valueminimum value


y = x + x – 5y = x + x – 5

a = 1, b= 1 c= -5a = 1, b= 1 c= -5

y-intercepty-intercept

y = 0 + 0 – 5y = 0 + 0 – 5

y = -5y = -5 Therefore the y–intercept Therefore the y–intercept isis

(0,-5)(0,-5)


y = x + x – 5y = x + x – 5

a = 1, b= 1 c= -5a = 1, b= 1 c= -5

x-interceptx-intercept

0 = x + x – 50 = x + x – 5

Using the discriminant, we Using the discriminant, we know that there are two real know that there are two real values of x because …values of x because …


y = x + x – 5y = x + x – 5

a = 1, b= 1 c= -5a = 1, b= 1 c= -5

√ √1- 4(1)(-5)1- 4(1)(-5)

is √21is √21

so using the quadratic so using the quadratic formula, we know that formula, we know that

X = -1/2 X = -1/2 ++ √21 √21


y = x + x – 5y = x + x – 5

a = 1, b= 1 c= -5a = 1, b= 1 c= -5

X = -1/2 X = -1/2 ++ √21 √21

which is approximatelywhich is approximately

-1/2 + 4.5 = 3.5 and-1/2 + 4.5 = 3.5 and

-1/2 – 4.5 = -5 -1/2 – 4.5 = -5


Now you are ready to sketch the Now you are ready to sketch the parabola, using all the information parabola, using all the information found earlierfound earlier

constructing parabolas from quadratics you need the following items to construct a parabola line of...

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