Quadratic Functions (20-1) Includes Vertex Form

Question 1

Rewrite x²+6x+8 in vertex form.

Answer 1

x² + 6x + 8
= (x² + 6x) + 8
Find the magic number by taking the coefficient in front of x which is 6 in this case, and divide it by 2, then square it. The magic number is (6÷2)² which equals 9.
= (x² + 6x + 9 - 9) + 8
= (x² + 6x + 9) - 9 + 8
= (x² + 6x + 9) - 1
When you find the magic number you divided the 6 by 2 and you got 3. Remember this number because it is useful here.
= (x + 3)² - 1

Use vertex form to find where the vertex is of the parabola.
(-3, -1)

(x+3)²-1

Question 2

Find the vertex of y = x² - 4x + 7.

Answer 2

To find the magic number we divide the coefficient B by 2 and then square it. Add and subtract that number.

y = x² - 4x + 7
y = (x² - 4x) + 7
y = (x² - 4x + 4 - 4)+ 7
y = (x² - 4x + 4) - 4 + 7
y = (x² - 4x + 4) + 3
y = (x - 2)² + 3

Vertex: (2,3)

Question 3

Find the axis of symmetry for y = 2(x - 5)² + 1.

Answer 3

x=5

The axis of symmetry is found by looking at horizontal translation.

Question 4

Determine if y = -x² + 2x - 3 opens up or down.

Answer 4

Opens down (a<0)

Question 5

Determine the maximum value of y=-2(x+1)²+8.

Answer 5

Maximum y=8

The maximum value is the y-coordinate of the vertex. It is a maximum here because the parabola opens down.

Question 6

Find the zeros of y = x² - 9

Answer 6

y=x²-9
y = (x + 3)(x - 3)
Sub in y = 0
0 = (x + 3)(x - 3)
0 = x + 3 0 = x - 3
x = -3. x = 3
x=±3

Question 7

Find the zeros of y = 2x² - 5x - 3.

Answer 7

y = 2x² - 5x - 3

Product -6
Sum -5

Two numbers are -6 and 1
Factor by decomposition because A is not 1.

y = 2x² - 5x - 3

Split B

y = 2x² - 6x + 1x - 3

Factor out in pairs

y = 2x(x - 3) + 1(x - 3)

Factor out the identical factor that just appeared.

y = 2x(x - 3) + 1(x - 3)

y = (x - 3)(2x + 1)

Set y = 0

0 = (x - 3)(2x + 1)

0 = x - 3. 0 = 2x + 1

x=3, x=-0.5

Question 8

Determine y-intercept of y = 3x² - 7x + 2.

Answer 8

Sub in x = 0

y-intercept = 2

Question 9

Find x-intercepts of y=(x-2)²-9.

Answer 9

y=(x-2)²-9

Sub in y = 0

0 = (x-2)² - 9

9 = (x-2)²
Square root both sides.

-+3 = x - 2

2 + 3 = x 2 - 3 = x

5 = x -1 = x

x=-1, x=5

Question 10

Sketch key points for y=(x-1)². State vertex and two symmetric points.

Answer 10

Vertex (1,0), Points (0,1) and (2,1)

Question 11

Solve x² + 8x + 15 = 0.

Answer 11

x²+8x+15=0

Product 15
Sum 8
Two numbers are 3 and 5
Since A = 1, don’t bother with decomposition

(x + 3)(x + 5) = 0

x=-3, x=-5

Question 12

Solve 2x² = 50.

Answer 12

2x² = 50
Divide both sides by 2
x² = 25
Square root both sides
x=5, x=-5

Question 13

Solve 6x² + 7x + 3 = 0.

It doesn’t factor nicely — what method should you use?

Answer 13

Use the quadratic formula, because the discriminant b² - 4ac = 49 - 72 = -23 is not a perfect square. Notice how there are no real solutions because the discriminant is negative. You can say there are no solutions, but it is possible to also report the imaginary solutions as below.
x = (-7 ± sqrt(-23)) / 12

Question 14

Solve 3x² + 2x - 5 = 0.

What method should you use, and what are the solutions?

Answer 14

Use factoring by decomposition.

Product -15
Sum 2
Two numbers are 5 and-3

3x² + 2x - 5 = 0
3x² + 5x - 3x - 5 = 0
x(3x + 5) -(3x + 5) = 0
(x - 1)(3x + 5) = 0
x = 1 or x = -5/3

You can also use other methods but since factoring is easily done, this is the best method.