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What is the value of h when the function is converted to vertex form?

Note: Vertex form is f(x)=a(x−h)2+k .

f(x)=x2+10x+35

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h =








What is the minimum value for h(x)=x2−16x+60?

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y =










What are the x-intercepts of the quadratic function?

f(x)=x2−3x−10



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Let ​ f(x)=x2+17x+72 ​ .



What are the zeros of the function?

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and









Let ​ f(x)=x2−8x+19 ​ .


What is the minimum value of the function?​

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Respuesta :

calculista
part 1) What is the value of h when the function is converted to vertex form?
f(x)=x²+10x+35
Group terms that contain the same variable
f(x)=(x²+10x)+35
Complete the square . Remember to balance the equation 
f(x)=(x²+10x+25)+35-25

Rewrite as perfect squares

f(x)=(x+5)²+10

(h,k) is (-5,10)

the answer Part 1) is 

h is -5


Part 2) What is the minimum value for h(x)=x²−16x+60?

h(x)=x²−16x+60

Group terms that contain the same variable

h(x)=(x²−16x)+60

Complete the square . Remember to balance the equation

h(x)=(x²−16x+64)+60 -64

Rewrite as perfect squares

h(x)=(x-8)²-4

(h,k) is the vertex-------> (8,-4)


the answer Part 2) is

the minimum value of h(x) is -4


Part 3)

What are the x-intercepts of the quadratic function?

f(x)=x²−3x−10

we know that the x intercepts is when y=0
x²−3x−10=0

Group terms that contain the same variable, and move the constant to the opposite side of the equation

(x²−3x)=10
Complete the square. Remember to balance the equation by adding the same constants to each side
(x²−3x+2.25)=10+2.25

Rewrite as perfect squares

(x-1.5)²=12.25---------> (+/-)[x-1.5]=3.5
(+)[x-1.5]=3.5-------> x1=5
(-)[x-1.5]=3.5------> x2=-2

the answer Part 3) is 
the x intercepts are
 x=5
x=-2

Part 4) Let ​ f(x)=x²+17x+72 ​ .

What are the zeros of the function?
x²+17x+72=0

Group terms that contain the same variable, and move the constant to the opposite side of the equation

(x²+17x)=-72
Complete the square. Remember to balance the equation by adding the same constants to each side
(x²+17x+72.25)=-72+72.25

Rewrite as perfect squares

(x+8.5)²=0.25-----------> (+/-)[x+8.5]=0.5

(+)[x+8.5]=0.5-----> x1=-8

(-)[x+8.5]=0.5-----> x2=-9

the answer part 4) is 

x=-8

x=-9

Part 5) Let ​ f(x)=x2−8x+19 ​ .

What is the minimum value of the function?
 f(x)=x²−8x+19
Group terms that contain the same variable
f(x)=(x²−8x)+19
Complete the square. Remember to balance the equation
f(x)=(x²−8x+16)+19-16

Rewrite as perfect squares

f(x)=(x-4)²+3

the vertex is the point (4,3)

the answer Part 5) is 

the minimum value of the function is 3 


 

The leading coefficient of the quadratic function determines if it has a

minimum or maximum value.

Correct responses;

  1. h = -5
  2. The minimum value is h(8) = -4
  3. The x-intercept are (5, 0) and (-2, 0)
  4. The zeros of the function are; x = -8, and x = -9
  5. The minimum value of the function is f(3) = 3

Method by which the above values are found;

(1) The vertex form of a parabola is; f(x) = a·(x - h)² + k

Where;

(h, k) = The coordinates of the vertex

The general form of the quadratic function is; y = a·x² + b·x + c

[tex]\displaystyle At \ the \ vertex, \ x = h = \mathbf{\frac{-b}{2 \cdot a}}[/tex]

Which gives;

[tex]\displaystyle h = \frac{-10}{2 \times 1} = -5[/tex]

  • h = -5

(2) At the minimum value, we have;

[tex]\displaystyle x = \mathbf{\frac{- (-16)}{2 \times 1}} = 8[/tex]

  • The minimum value is therefore; h(8) = 8² - 16×8 + 60 = -4

(3) The given function is; f(x) = x² - 3·x - 10

At the x-intercept, f(x) = 0

Which gives;

f(x) = 0 = x² - 3·x - 10 = (x - 5)·(x + 2)

(x - 5)·(x + 2) = 0

At the x-intercept, f(x) = 0, x = 5, x = -2, which gives;

  • The x-intercept are; (5, 0), and (-2, 0)

(4) The given function is; f(x) = x² + 17·x + 72

The zeros are the point where the value of the function is 0, which are

given as follows;

f(x) = 0 = x² + 17·x + 72 = (x + 8)·(x + 9)

(x + 8)·(x + 9) = 0

The values of x that gives the zeros of f(x) are; x = -8, and x = -9

  • The zeros of the function are; x = -8, and x = -9

(5) The leading coefficient of f(x) = x² - 8·x + 19 is positive, therefore, the function has a minimum value

At the minimum value, x = [tex]\displaystyle \mathbf{\frac{-(-8)}{2 \times 1}} = 4[/tex]

The minimum value is the value of f(x) at x = 4, which gives;

f(4) = 4² - 8×4 + 19 = 3

  • The minimum value of the function is f(3) = 3

Learn more about quadratic functions here:

https://brainly.com/question/1478218

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