Question

select all the correct locations on the image. identify which functions have complex roots by selecting the function names on the provided coordinate plane.

Explanation:

To determine which functions have complex roots, we use the fact that a real - valued function (defined on the real numbers) has a real root when its graph intersects the \(x\) - axis (i.e., when \(y = 0\) for some real \(x\)). If a function's graph does not intersect the \(x\) - axis, then the equation \(f(x)=0\) has no real solutions, and thus the roots of the equation \(f(x) = 0\) must be complex (since non - real roots of polynomial equations with real coefficients come in conjugate pairs).

Analyzing each function:

• Function \(a\): The graph of function \(a\) is a parabola that opens upwards and is entirely above the \(x\) - axis. So, it never intersects the \(x\) - axis. This means the equation \(y=a(x)\) (or \(a(x)=0\)) has no real roots, so it has complex roots.
• Function \(b\): The graph of function \(b\) is a parabola that opens upwards and is entirely above the \(x\) - axis. It does not intersect the \(x\) - axis, so the equation \(b(x)=0\) has no real roots and thus has complex roots.
• Function \(c\): The graph of function \(c\) intersects the \(x\) - axis (we can see that it crosses the \(x\) - axis at some real \(x\) - values). So, it has real roots.
• Function \(d\): The graph of function \(d\) is a parabola that opens downwards and is entirely below the \(x\) - axis. It does not intersect the \(x\) - axis, so the equation \(d(x)=0\) has no real roots and thus has complex roots.
• Function \(e\): The graph of function \(e\) (a straight line, likely a linear function) intersects the \(x\) - axis, so it has real roots.
• Function \(f\): The graph of function \(f\) is a parabola that opens downwards and is entirely below the \(x\) - axis. It does not intersect the \(x\) - axis, so the equation \(f(x)=0\) has no real roots and thus has complex roots.

Answer:

The functions with complex roots are the ones corresponding to the labels \(a\), \(b\), \(d\), and \(f\). So we select the locations labeled \(a\), \(b\), \(d\), and \(f\) on the coordinate plane.