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

Solution Steps

The given quadratic function is

\[ f(x) = x^2 + 14x - 29 \]

From this, we identify the coefficients:

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

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

\[ h = -\frac{b}{2a} \]

Substituting the values of \( b \) and \( a \):

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

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

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

Calculating this step-by-step:

\[ = 49 - 98 - 29 = -78.0 \]

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

\[ f(x) = (x - h)^2 + k = (x - (-7.0))^2 - 78.0 \]

This simplifies to:

\[ f(x) = (x + 7.0)^2 - 78.0 \]

Final Answer