Question

from the factored form of h(x), what are the x-intercepts of the roller coaster height graph?
x=0,2,3 with x=2 double root
x=0,1,3 with x=1 double root
x=0,2,3 with multiplicity one each
x=0,1,2 with multiplicity one each

Explanation:

To determine the x - intercepts of a function from its factored form, we use the property that if a function \(H(x)\) is in factored form, say \(H(x)=a(x - r_1)(x - r_2)\cdots(x - r_n)\) (where \(a\) is a non - zero constant and \(r_i\) are the roots), then the x - intercepts are the values of \(x\) for which \(H(x) = 0\). This occurs when \(x=r_i\) for each root \(r_i\).

Analyzing the options:

• For a root with multiplicity one, it means the factor \((x - r)\) appears once in the factored form of the function. For a double root, the factor \((x - r)\) appears twice (i.e., \((x - r)^2\) is a factor of the function).
• Let's assume the factored form of \(H(x)\) (even though it's not fully given, we can analyze the options based on the concept of roots and their multiplicities). If the roots are \(x = 0\), \(x=2\), \(x = 3\) and each has multiplicity one, it means that the factors \((x-0)=x\), \((x - 2)\) and \((x - 3)\) each appear once in the factored form of \(H(x)\). So the x - intercepts are \(x = 0\), \(x=2\) and \(x = 3\) with each having a multiplicity of one.

Now let's analyze the other options:

• Option with \(x = 0,2,3\) and \(x = 2\) as a double root: If \(x = 2\) is a double root, the graph of the function will touch the x - axis at \(x = 2\) and cross at the other roots. But since we are to find the x - intercepts from the factored form (assuming a standard case where we don't have a double root for \(x = 2\) unless specified), this is less likely.
• Option with \(x=0,1,3\) and \(x = 1\) as a double root: There is no indication that \(x = 1\) should be a root, so this option is incorrect.
• Option with \(x=0,1,2\) and each with multiplicity one: There is no indication that \(x = 1\) should be a root, so this option is incorrect.

The x - intercepts of a function in factored form are the roots (values of \(x\) that make the function zero). A root with multiplicity one means its corresponding factor appears once in the factored form. Among the options, the roots \(x = 0\), \(x = 2\), \(x=3\) with multiplicity one each is consistent with the general concept of finding x - intercepts from a factored function (assuming no double - root indication for \(x = 2\) and no presence of \(x = 1\) as a root).

Answer:

x = 0, 2, 3 with multiplicity one each (the option in the orange box: "x=0,2,3 with multiplicity one each")