Math  /  Algebra

Question5. Which of the following quadratic graph open upward? A. y=(x+2)(3x)y=(x+2)(3-x) B. y=9x+124x2y=9 x+12-4 x^{2} C. y=x224y=-x^{2}-24 D. y=(2x+1)(x+5)y=(2 x+1)(x+5)
6. What is the vertex of the quadratic graph y=(x+3)28y=-(x+3)^{2}-8 A. (3,8)(-3,8) B. (3,8)(-3,-8) C. (3,8)(3,-8) D. (3,8)(3,8)
7. What is the yy-intercept of the quadratic graph y=3(x3)2+7y=3(x-3)^{2}+7 A. 34 B. 7 C. 27 D. 3

Studdy Solution

STEP 1

What is this asking? We need to figure out which parabola opens upwards, find the vertex of a parabola, and find the y-intercept of another parabola. Watch out! Remember, the sign of the coefficient of the x2x^2 term tells us whether a parabola opens upwards or downwards.
Also, be careful with the signs when finding the vertex from vertex form!

STEP 2

1. Upward or Downward Parabolas
2. Vertex Location
3. Y-intercept Calculation

STEP 3

Alright, let's tackle the first question!
We're looking for a parabola that opens upwards, which means the coefficient of the x2x^2 term must be **positive**.

STEP 4

Let's expand option A: y=(x+2)(3x)=3xx2+62x=x2+x+6y = (x+2)(3-x) = 3x - x^2 + 6 - 2x = -x^2 + x + 6.
See that x2-x^2?
The coefficient of x2x^2 is 1-1, which is **negative**, so this parabola opens downwards.

STEP 5

Next, option B: y=9x+124x2=4x2+9x+12y = 9x + 12 - 4x^2 = -4x^2 + 9x + 12.
Here, the coefficient of x2x^2 is 4-4, which is **negative**.
Downwards again!

STEP 6

Option C: y=x224y = -x^2 - 24.
The coefficient of x2x^2 is 1-1, **negative**!
This one opens downwards too.

STEP 7

Finally, option D: y=(2x+1)(x+5)=2x2+10x+x+5=2x2+11x+5y = (2x+1)(x+5) = 2x^2 + 10x + x + 5 = 2x^2 + 11x + 5.
The coefficient of x2x^2 is 22, which is **positive**!
This parabola opens upwards.
So, the answer to question 5 is **D**.

STEP 8

The equation is already in vertex form: y=(x+3)28y = -(x+3)^2 - 8.
Remember, the vertex form is y=a(xh)2+ky = a(x-h)^2 + k, where (h,k)(h, k) is the **vertex**.

STEP 9

In our equation, we can rewrite it as y=(x(3))2+(8)y = -(x - (-3))^2 + (-8).
So, h=3h = -3 and k=8k = -8.

STEP 10

Therefore, the vertex is (3,8)(-3, -8).
The answer to question 6 is **B**.

STEP 11

To find the y-intercept, we set x=0x = 0 in the equation y=3(x3)2+7y = 3(x-3)^2 + 7.

STEP 12

Substituting x=0x = 0, we get y=3(03)2+7=3(3)2+7=3(9)+7=27+7=34y = 3(0-3)^2 + 7 = 3(-3)^2 + 7 = 3(9) + 7 = 27 + 7 = 34.

STEP 13

So, the y-intercept is 3434.
The answer to question 7 is **A**.

STEP 14

5. **D**
6. **B**
7. **A**

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