VERTEX OF A PARABOLA WORKSHEET

Problems 1-8 : Find the vertex of the parabola.

Problem 1 :

y = x² - 2x - 5

Problem 2 :

y = -x² - 14x - 59

Problem 3 :

y = x² + 4x

Problem 4 :

x = y² - 2y - 5

Problem 5 :

x = -y² - 14y - 59

Problem 6 :

x = y² + 4y

Problem 7 :

y = -3(x + 2)(x - 6)

Problem 8 :

y = 2x(x - 3)

Problems 8-10 : Write the equation of the parabola in vertex form and find the vertex from it.

Problem 9 :

y = x² - 6x + 10

Problem 10 :

x = -y² - 12y - 40

1. Answer :

Comparing y = ax² + bx + c and y = x² - 2x - 5,

a = 1, b = -2 and c = -5

x-coordinate of the vertex :

x = -b/2a

Substitute a = 1 and b = -2.

x = -(-2)/2(1)

x = 2/2

x = 1

y-coordinate of the vertex :

Substitute x = 1 in y = x² - 2x - 5.

y = 1² - 2(1) - 5

y = 1 - 2 - 5

y = -6

Vertex of the parabola is (1, -6).

2. Answer :

Comparing y = ax² + bx + c and y = -x² - 14x - 59,

a = -1, b = -14 and c = -59

x-coordinate of the vertex :

x = -b/2a

Substitute a = -1 and b = -14.

x = -(-14)/2(-1)

x = 14/(-2)

x = -7

y-coordinate of the vertex :

Substitute x = -7 in y = -x² - 14x - 59.

y = -(-7)² - 14(-7) - 59

y = -49 + 98 - 59

y = -10

Vertex of the parabola is (-7, -10).

3. Answer :

Comparing y = ax² + bx + c and y = x² + 4x,

a = 1, b = 4 and c = 0

x-coordinate of the vertex :

x = -b/2a

Substitute a = 1 and b = 4.

x = -4/2(1)

x = -4/2

x = -2

y-coordinate of the vertex :

Substitute x = -2 in y = x² + 4x.

y = (-2)² + 4(-2)

y = 4 - 8

y = -4

Vertex of the parabola is (-2, -4).

4. Answer :

Comparing x = ay² + by + c and x = y² - 2y - 5,

a = 1, b = -2 and c = -5

y-coordinate of the vertex :

y = -b/2a

Substitute a = 1 and b = -2.

y = -(-2)/2(1)

y = 2/2

y = 1

x-coordinate of the vertex :

Substitute y = 1 in x = y² - 2y - 5.

x = 1² - 2(1) - 5

x = 1 - 2 - 5

x = -6

Vertex of the parabola is (-6, 1).

5. Answer :

Comparing x = ay² + by + c and x = -y² - 14y - 59,

a = -1, b = -14 and c = -59

x-coordinate of the vertex :

y = -b/2a

Substitute a = -1 and b = -14.

y = -(-14)/2(-1)

y = 14/(-2)

y = -7

y-coordinate of the vertex :

Substitute y = -7 in x = -y² - 14y - 59.

x = -(-7)² - 14(-7) - 59

x = -49 + 98 - 59

x = -10

Vertex of the parabola is (-10, -7).

6. Answer :

Comparing x = ay² + by + c and x = y² + 4y,

a = 1, b = 4 and c = 0

x-coordinate of the vertex :

y = -b/2a

Substitute a = 1 and b = 4.

y = -4/2(1)

y = -4/2

y = -2

y-coordinate of the vertex :

Substitute y = -2 in x = y² + 4y.

x = (-2)² + 4(-2)

x = 4 - 8

x = -4

Vertex of the parabola is (-4, -2).

7. Answer :

y = -3(x + 2)(x - 6)

Substitute y = 0 to find the x-intercepts of the parabola above.

-3(x + 2)(x - 6) = 0

Divide both sides by -3.

-3(x + 2)(x - 6) = 0

x + 2 = 0  or  x - 6 = 0

x = -2  or  x = 6

x-coordinate of the vertex :

x = (-2 + 6)/2

x = 4/2

x = 2

y-coordinate of the vertex :

Substitute x = 2 in y = -3(x + 2)(x - 6).

y = -3(2 + 2)(2 - 6)

y = -3(4)(-4)

y = 48

Vertex of the parabola is (2, 48).

8. Answer :

y = 2x(x - 3)

To find the x-intercepts of the parabola above, substitute y = 0.

2x(x - 3) = 0

Divide both sides by 2.

x(x - 3) = 0

x = 0  or  x = 3

x-coordinate of the vertex :

x = (0 + 3)/2

x = 3/2

x = 1.5

y-coordinate of the vertex :

Substitute x = 1.5 in y = 2x(x - 3).

y = 2(1.5)(1.5 - 3)

y = 2(1.5)(-1.5)

y = -4.5

Vertex of the parabola is (1.5, -4.5).

9. Answer :

y = x² - 6x + 10

y = x² - 2(x)(3) + 3² - 3² + 10

Using the identity (a - b)² = a² - 2ab + b²,

y = (x - 3)² - 3² + 10

y = (x - 3)² - 9 + 10

y = (x - 3)² + 1

Comparing y = a(x - h)² + k and y = (x - 3)² + 1,

h = 3 and k = 1

Vertex of the parabola :

(h, k) = (3, 1)

10. Answer :

x = -y² - 12y - 40

x = -1(y² + 12y) - 40

x = -1[y² + 2(y)(6) + 6² - 6²] - 40

Using the identity (a + b)² = a² + 2ab + b²,

x = -1[(y + 6)² - 6²] - 40

x = -1[(y + 6)² - 36] - 40

x = -1(y + 6)² + 36 - 40

x = -1(y + 6)² - 4

The vertex form equation x = -1(y + 6)² - 10 can be written as

x = -1[y - (-6)]² + (-4)

Comparing the above equation and x = a(y - k)² + h,

h = -4 and k = -6

Vertex of the parabola :

(h, k) = (-4, -6)

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