Questions: Write the quadratic function in standard form. h(x)=x^2-14 x+49 h(x)= Sketch its graph. Identify the vertex, axis of symmetry, and x-intercept(s), (If an answer does not exist, enter DNE.) vertex (x, h(x))=(sqrt( )) axis of symmetry x-intercept (x, h(x))=(sqrt( ))

Solution Steps

The given quadratic function is \( h(x) = -x^2 - 14x + 49 \). This is already in standard form, which is \( ax^2 + bx + c \).

The vertex form of a quadratic function is \( h(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex. To find the vertex, use the formula \( h = -\frac{b}{2a} \).

For the given function \( h(x) = -x^2 - 14x + 49 \):

• \( a = -1 \)
• \( b = -14 \)

Calculate \( h \): \[ h = -\frac{-14}{2(-1)} = \frac{14}{-2} = -7 \]

Now, substitute \( x = -7 \) back into the function to find \( k \): \[ h(-7) = -(-7)^2 - 14(-7) + 49 \] \[ h(-7) = -49 + 98 + 49 \] \[ h(-7) = 98 \]

So, the vertex is \((-7, 98)\).

The axis of symmetry for a quadratic function \( ax^2 + bx + c \) is given by \( x = -\frac{b}{2a} \).

From the previous step, we found: \[ x = -7 \]

Final Answer