Questions: Write the quadratic function f(x)=x^2+14x-29 in the form f(x)=(x-h)^2+k f(x)=

Write the quadratic function f(x)=x^2+14x-29 in the form f(x)=(x-h)^2+k

f(x)=
Transcript text: Write the quadratic function $f(x)=x^{2}+14 x-29$ in the form $f(x)=(x-h)^{2}+k$ \[ f(x)= \]
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Solution

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Solution Steps

Step 1: Identify the Coefficients

The given quadratic function is

f(x)=x2+14x29 f(x) = x^2 + 14x - 29

From this, we identify the coefficients:

a=1,b=14,c=29 a = 1, \quad b = 14, \quad c = -29

Step 2: Calculate h h

To find the x-coordinate of the vertex h h , we use the formula

h=b2a h = -\frac{b}{2a}

Substituting the values of b b and a a :

h=1421=7.0 h = -\frac{14}{2 \cdot 1} = -7.0

Step 3: Calculate k k

Next, we evaluate the original function at x=h x = h to find the y-coordinate k k :

k=f(h)=f(7)=1(7)2+14(7)29 k = f(h) = f(-7) = 1 \cdot (-7)^2 + 14 \cdot (-7) - 29

Calculating this step-by-step:

=499829=78.0 = 49 - 98 - 29 = -78.0

Step 4: Write the Function in Vertex Form

Now that we have h h and k k , we can express the function in vertex form:

f(x)=(xh)2+k=(x(7.0))278.0 f(x) = (x - h)^2 + k = (x - (-7.0))^2 - 78.0

This simplifies to:

f(x)=(x+7.0)278.0 f(x) = (x + 7.0)^2 - 78.0

Final Answer

The quadratic function in vertex form is

f(x)=(x+7)278 \boxed{f(x) = (x + 7)^2 - 78}

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