Questions: Let h(x)=7 x^2. Determine the root(s) of h, as well as the x-value of the vertex of the graph of h.

Transcript text: Let $h(x)=7 x^{2}$. Determine the $\operatorname{root}(\mathrm{s})$ of $h$, as well as the $x$-value of the vertex of the graph of $h$.

Solution

Solution Steps

To solve the given problem, we need to find the roots and the x-value of the vertex of the quadratic function $ h(x) = 7x^2 $.

Roots: The roots of the function are the values of $ x $ for which $ h(x) = 0 $. For a quadratic function $ ax^2 + bx + c = 0 $, the roots can be found using the quadratic formula $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $. In this case, since $ b = 0 $ and $ c = 0 $, the roots are simply $ x = 0 $.
Vertex: The x-value of the vertex of a quadratic function $ ax^2 + bx + c $ is given by $ x = -\frac{b}{2a} $. For the function $ h(x) = 7x^2 $, $ b = 0 $, so the x-value of the vertex is $ x = 0 $.

Step 1: Finding the Roots

To find the roots of the function $ h(x) = 7x^2 $, we set the equation equal to zero: \[ 7x^2 = 0 \] Dividing both sides by 7 gives: \[ x^2 = 0 \] Taking the square root of both sides results in: \[ x = 0 \] Thus, the root of the function is $ x = 0 $.

Step 2: Finding the x-value of the Vertex

The x-value of the vertex for a quadratic function $ ax^2 + bx + c $ is given by the formula: \[ x = -\frac{b}{2a} \] For our function, $ a = 7 $ and $ b = 0 $. Substituting these values into the formula gives: \[ x = -\frac{0}{2 \cdot 7} = 0 \] Therefore, the x-value of the vertex is $ x = 0 $.