Question

What’s is part A?? .....

Answer 1

Equation A matches Graph 3, because the parabola is opened upward and intersected the x-axis at points (2 , 0) and (4 , 0)

Equation B matches Graph 4, because the parabola is opened upward, intersected the x-axis at point (-8 , 0) , (6 , 0) and the y-axis at point (0 , -48)

Equation C matches Graph 1, because the parabola is above the x-axis

Equation D matches Graph 2, because the parabola is opened downward

Explanation:

The quadratic function f(x) = ax² + bx + c is represented graphically by a parabola which has a vertex point (h , k)

The vertex form of the quadratic function is f(x) = (x - h)² + k

• The parabola is opened upward if a is positive or downward if a is negative
• The parabola intersected the y-axis at c
• The parabola intersected the x-axis at the zeroes of the function
• If k the y-coordinate of the vertex of the parabola is greater than 0, then the parabola never intersects the x-axis

Equation A

∵ y = x² - 6x + 8

∵ a = 1

- By using the 1st note above

∴ The graph of the equation is a parabola opened upward

- Lets find its zeros (Put y = 0)

∵ x² - 6x + 8 = 0

- Factorize it into two factors

∴ (x - 2)(x - 4) = 0

- Equate each factor by 0

∵ x - 2 = 0 ⇒ add 2 to both sides

∴ x = 2

∵ x - 4 = 0 ⇒ add 4 to both sides

∴ x = 4

∴ Its zeroes are 2 and 4

- That means the parabola intersects the x-axis at 2 and 4

∴ The graph of equation A is figure 3

Equation A matches Graph 3, because the parabola is opened upward and intersected the x-axis at points (2 , 0) and (4 , 0)

Equation B

∵ y = (x - 6)(x + 8)

- Equate each factor by 0

∵ x - 6 = 0 ⇒ add 6 to both sides

∴ x = 6

∵ x + 8 = 0 ⇒ subtract 8 from both sides

∴ x = -8

∴ Its zeroes are 2 and 4

- That means the parabola intersects the x-axis at points (-8 , 0)

and (6 , 0)

∵ (x - 6)(x + 8) = x² + 8x - 6x - 48

- Add like terms

∴ (x - 6)(x + 8) = x² + 2x - 48

∴ y = x² + 2x - 48

- Find the value of a and c

∴ a = 1 and c = -48

∵ a is positive

∴ The parabola is opened upward

∵ c = -48

∴ The parabola intersects the y-axis at point (0 , -48)

∴ The graph of equation B is figure 4

Equation B matches Graph 4, because the parabola is opened upward intersected the x-axis at point (-8 , 0) , (6 , 0) and the y-axis at point (0 , -48)

Equation C

∵ y = (x - 6)² + 8

- The equation is in the vertex form

∴ The coordinates of the vertex point are h = 6 and k = 8

∴ The vertex of the parabola is (6 , 8)

- By using the 4th note above

∵ k > 0

∴ The parabola does not intersect the x-axis

∴ the parabola is above the x-axis

∴ The graph of equation C is figure 1

Equation C matches Graph 1, because the parabola is above the x-axis

Equation D

∵ y = -(x + 8)(x - 6)

(x + 8)(x - 6) = x² - 6x + 8x - 48

- Add like terms

∴ -(x - 6)(x + 8) = -(x² + 2x - 48)

- Multiply each term by (-)

∴ -(x - 6)(x + 8) = - x² - 2x + 48

∴ y = -x² - 2x + 48

- Find the value of a

∴ a = -1

- Use the 1st rule above

∵ a < 0

∴ The parabola is opened downward

∴ The graph of equation D is figure 2

Equation D matches Graph 2, because the parabola is opened downward

Answer 2

A. 3

B. 4

C. 1

D. 2

Explanation:

Consider all equations:

A.
`y=x^2 -6x+8`

This is the equation of parabola with vertex at point

`x_v=(-b)/(2a)=(-(-6))/(2\cdot 1)=3\\ \\y_v=3^2-6\cdot 3+8=9-18+8=-1`

The y-intercept is at point

`x=0\\ \\y=0^2-6\cdot 0+8=8`

Since

`x^2 -6x+8=x^2-4x-2x+8=x(x-4)-2(x-4)=(x-2)(x-4)`,

the x-intercepts are at points (2,0) and (4,0)

The leading coefficient is 1 > 0, then the parabola opens upwards.

Hence, the graph of this parabola is 3.

B.
`y=(x-6)(x+8)`

This parabola has two x-intercepts at points (6,0) and (-8,0).

The leading coefficient is 1 > 0, then the parabola opens upwards.

The only possible choice is parabola 4.

C.
`y=(x-6)^2+8`

This parabola has the vertex at point (6,8), opens upwards, therefore does not intersect the x-axis.

The graph of this parabola is 1.

D.
`y=-(x+8)(x-6)`

This parabola has two x-intercepts at points (6,0) and (-8,0).

The leading coefficient is -1 > 0, then the parabola opens downwards.

The only possible choice is parabola 2.