NEed help simplifying!?

Question

trying to prepare for a test when I get back to school and I cant remember how to do this :(




Find the following, given the equation y = x2 + 4x - 21

a. Equation in general form,y = a(x – h)2 + k

b. Coordinate of the vertex

c. Y-intercept

d. X-intercept(s)

Answer 1

Hi,

Find the following, given the equation y = x² + 4x - 21

a. Equation in general form,y = a(x – h)2 + k
y = x² + 4x - 21
½*4 = 2 and 2² = 4 so add 4 to complete the square
Then subtract 4 on the end to balance that change.

y = x² + 4x + 4 - 21 - 4 Factor the perfect square trinomial.
y = (x² + 4x + 4) - 21 - 4
y = (x + 2)² - 21 - 4 <= Combine these numbers.
y = (x + 2)² - 25 <== ANSWER

b. Coordinate of the vertex
This equation is of the form y = a(x - h)² + k where (h,k) is the vertex. So the vertex is (-2,-25). <== ANSWER

c. Y-intercept
The y intercept is the constant in the original equation, so the Y-intercept is (0,-21). <== ANSWER

d. X-intercept(s)
x² - 4x - 21 = 0 factors into:
(x - 7)(x + 3) = 0
This solves to:
x - 7 = 0 and x + 3 = 0
x = 7 and x = -3
The x intercepts are (7,0) and (-3,0). <== ANSWER

I hope that helps!! :-)

Answer 2

Pi R-squared got the first CORRECT answer .. the answers above him have mistakes in them .........

(mine are right too, but I took a slightly different approach)

- - -end edit - - -


starting from "easiest"

c) Y-intercept ..
let x = 0 (this points to y-axis) ... notice that
y = 21 becomes your answer

d) x-intercept ... factor the equation
y = x2 + 4x - 21
y = (x-3 ) * (x+7)
y = 0 when either (x-3) or (x+7) equals 0, so
x - intercepts = 3 and -7

a) "general form" ... this requires "completing the square"

look at the first two terms in x2 + 4x - 21
x^2 + 4x .........

what would the 3rd term
note that (x-h)^2 = x^2 - 2xh + h^2
so we can see that
-2xh = 4x ........... meaning that
-2h = 4
h = -2

but ...
(x+2)^2 = x^2 +4x + 4 ........... so we need to subtract 25 toget our origninal eqn


y = x2 + 4x - 21 = (x + 2)^2 -25
y = 1*(x - (-2)) - 25 <========== answer to "a"

a = 1
h = -2
k = -25

b) Coordinate of the vertex [memorize this one]
vertex x-coordinate = -b / 2a when
y = ax^2 + bx + c

put in this x-value and solve for y.

here:
-b / 2a = -4 / 2 = -2 for x-coordinate

"plug in" this -2 and get
y = x2 + 4x - 21
y = (-2)^2 + 4(-2) - 21
y = 4 - 8 - 21
y = -25

vertex = ( -2, -25) <====== answer "b"

NOTICE: the vertetx x-coordinate is the midpoint ("average")
betweeen your x-intercepts of 3 and -7
(-7 + 3) / 2 = -4 / 2 = -2

a bit of "symmetry" . kinda cool

good luck ........ STUDY this stuff and do your homework
it's all learnable

Answer 3

y = x^2 + 4x -21

a)

complete the square by adding 4 on both sides

y +4 = x^2 + 4x + 4 - 21

y + 4 = (x+2)^2 -17

y = (x+2)^2 - 21 , where a = 1 , k = -21

b)

coordinates of vertex = (-2, -21)

c)

y intercept

put x = 0

0 + 0 -21 = -21

so y-intercept = -21

d)

x-intercept

put y = 0

x^2 + 4x - 21 = 0

(x+7)(x-3) = 0

so x = -7 or 3

so x-intercepts are -7 and 3

Answer 4

y = x^2 + 4x + 4 - 21 + 4
y = (x+2)^2 - 17
vertex = (h,k) = (-2,-17)
y intercept, when x = 0, y = -21
x intercept, solve x^2 + 4x - 21 = 0
(x+7)(x-3)= 0 x=-7 or 3

Answer 5

Cross multiplication (: