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