Constructing Parabolas from Quadratics You need the following items to construct a parabola Line of Symmetry (axis of symmetry) Line of Symmetry (axis of symmetry) Vertex Vertex Direction of opening Direction of opening Min/max vertex value Min/max vertex value Y-intercept Y-intercept X-intercepts X-intercepts Plot all these points and draw

Constructing Parabolas from Quadratics When you identify you a, b, & c, plug in the values to each part Line of Symmetry (axis of symmetry) Line of Symmetry (axis of symmetry) x = -b x = -b 2a 2a

Constructing Parabolas from Quadratics Vertex Vertex Plug in your value of x from the line of symmetry and find your value of y Plug in your value of x from the line of symmetry and find your value of y

Constructing Parabolas from Quadratics Direction of opening Direction of opening If a > 0 If a > 0 Then the parabola opens up Then the parabola opens up If a < 0 If a < 0 Then the parabola opens down Then the parabola opens down

Constructing Parabolas from Quadratics Minimum/Maximum vertex value Minimum/Maximum vertex value If a > 0 If a > 0 Then the vertex is the minimum value Then the vertex is the minimum value If a < 0 If a < 0 Then the vertex is the maximum value Then the vertex is the maximum value

Constructing Parabolas from Quadratics Y-intercept Y-intercept To find the y-intercept, To find the y-intercept, Set x = 0 and Set x = 0 and solve of y solve of y

Constructing Parabolas from Quadratics x-intercepts x-intercepts To find the x-intercepts (don’t forget there can be 2 x- intercepts) To find the x-intercepts (don’t forget there can be 2 x- intercepts) Set y = 0 and Set y = 0 and solve of x solve of x

Constructing Parabolas from Quadratics Remember the 3 ways to find the x-intercepts Factoring Factoring Completing the square Completing the square Using the quadratic formulaUsing the quadratic formula

Constructing Parabolas from Quadratics Remember to determine if there are REAL values of x, you use the discriminant √ b-4ac √ b-4ac If discriminant > 0, there are 2 REAL values of x there are 2 REAL values of x If discriminant = 0, there is 1 REAL value of x there is 1 REAL value of x If discriminant < 0, there are NO REAL valuesof x there are NO REAL valuesof x

Constructing Parabolas from Quadratics Example: Construct the parabola for the following quadratic equation Construct the parabola for the following quadratic equation y = x + x - 5 y = x + x - 5

Constructing Parabolas from Quadratics y = x + x – 5 y = x + x – 5 a = 1, b= 1 c= -5 a = 1, b= 1 c= -5 Line of symmetry Line of symmetry x = -1 x = -1/2 x = -1 x = -1/2 2(1) 2(1)

Constructing Parabolas from Quadratics y = x + x – 5 y = x + x – 5 a = 1, b= 1 c= -5 a = 1, b= 1 c= -5 Vertex Vertex y = (1/2) - ½ - 5 y = (1/2) - ½ - 5 y = ¼ - ½ - 5 y = ¼ - ½ - 5 y = ¼ - 2/4 – 20/4 y = ¼ - 2/4 – 20/4 y = -21/4 y = -21/4 Therefore, the vertex is (-1/2, -21/4) Therefore, the vertex is (-1/2, -21/4)

Constructing Parabolas from Quadratics y = x + x – 5 y = x + x – 5 a = 1, b= 1 c= -5 a = 1, b= 1 c= -5 Direction of opening Direction of opening a > 0 a > 0 Therefore, the parabola opens up Therefore, the parabola opens up

Constructing Parabolas from Quadratics y = x + x – 5 y = x + x – 5 a = 1, b= 1 c= -5 a = 1, b= 1 c= -5 Min/Mix Vertex Value Min/Mix Vertex Value a > 0 a > 0 Therefore, the vertex value is the minimum value Therefore, the vertex value is the minimum value

Constructing Parabolas from Quadratics y = x + x – 5 y = x + x – 5 a = 1, b= 1 c= -5 a = 1, b= 1 c= -5 y-intercept y-intercept y = 0 + 0 – 5 y = 0 + 0 – 5 y = -5 y = -5 Therefore the y–intercept is Therefore the y–intercept is (0,-5) (0,-5)

Constructing Parabolas from Quadratics y = x + x – 5 y = x + x – 5 a = 1, b= 1 c= -5 a = 1, b= 1 c= -5 x-intercept x-intercept 0 = x + x – 5 0 = x + x – 5 Using the discriminant, we know that there are two real values of x because … Using the discriminant, we know that there are two real values of x because …

Constructing Parabolas from Quadratics y = x + x – 5 y = x + x – 5 a = 1, b= 1 c= -5 a = 1, b= 1 c= -5 √1- 4(1)(-5) √1- 4(1)(-5) is √21 is √21 so using the quadratic formula, we know that so using the quadratic formula, we know that X = -1/2 + √21 X = -1/2 + √21

Constructing Parabolas from Quadratics y = x + x – 5 y = x + x – 5 a = 1, b= 1 c= -5 a = 1, b= 1 c= -5 X = -1/2 + √21 X = -1/2 + √21 which is approximately which is approximately -1/ = 3.5 and -1/ = 3.5 and -1/2 – 4.5 = -5 -1/2 – 4.5 = -5

Constructing Parabolas from Quadratics Now you are ready to sketch the parabola, using all the information found earlier Now you are ready to sketch the parabola, using all the information found earlier