Graph the following quadratics and state its vertex, intercepts and domain & range.
$h\left(x\right)=\frac{1}{2}{x}^2+10x-5$

• A. Vertex: $\(\left\)(-10,-55\(\right\))$; $x$-intercepts: $\(\left\)(0.5,0\(\right\)),\(\left\)(-20.5,0\(\right\))$; $y$-intercept: $\(\left\)(0,-5\(\right\))$; Domain: $\(\left\)(-\(\infty\),\(\infty\]\right\))$; Range: $\(\left\[\lbrack\)-55,\(\infty\]\right\))$
• B. Vertex: $\(\left\)(10,55\(\right\))$; $x$-intercepts: $\(\left\)(-0.5,0\(\right\)),\(\left\)(20.5,0\(\right\))$; $y$-intercept: $\(\left\)(0,-5\(\right\))$; Domain: $\(\left\)(-\(\infty\),\(\infty\]\right\))$; Range: $\(\left\[\lbrack\)55,\(\infty\]\right\))$
• C. Vertex: $\(\left\)(-10,-55\(\right\))$; $x$-intercepts: $\(\left\)(0.5,0\(\right\)),\(\left\)(-20.5,0\(\right\))$; $y$-intercept: $\(\left\)(0,-5\(\right\))$; Domain: $\(\left\)(-\(\infty\),\(\infty\]\right\))$; Range: $\(\left\[\lbrack\)-55,\(\infty\]\right\))$
• D. Vertex: $\(\left\)(10,55\(\right\))$; $x$-intercepts: $\(\left\)(-0.5,0\(\right\)),\(\left\)(20.5,0\(\right\))$; $y$-intercept: $\(\left\)(0,-5\(\right\))$; Domain: $\(\left\)(-\(\infty\),\(\infty\]\right\))$; Range: $\(\left\[\lbrack\)55,\(\infty\]\right\))$

Answer: (C)

Verified step by step guidance

1. Identify the quadratic function given: \(h(x) = \frac{1}{2}x^2 + 10x - 5\).
2. Find the vertex using the vertex formula \(x = -\frac{b}{2a}\), where \(a = \frac{1}{2}\) and \(b = 10\). Substitute this \(x\) value back into \(h(x)\) to find the \(y\)-coordinate of the vertex.
3. Calculate the \(x\)-intercepts by setting \(h(x) = 0\) and solving the quadratic equation \(\frac{1}{2}x^2 + 10x - 5 = 0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
4. Find the \(y\)-intercept by evaluating \(h(0)\), which is simply the constant term in the quadratic expression.
5. State the domain and range: The domain of any quadratic function is all real numbers, \((-\infty, \infty)\). The range depends on the vertex; since \(a > 0\), the parabola opens upward, so the range is \([y_{vertex}, \infty)\) where \(y_{vertex}\) is the \(y\)-value of the vertex.