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:

Identify real roots from x-intercepts

The real roots of a function correspond to the points where its graph intersects or touches the \(x\)-axis.

• Curve \(a\) (light blue, opening upwards) crosses the \(x\)-axis at \(x = -4\) and \(x = 1\). Thus, it has real roots.
• Curve \(c\) (dark blue/purple, opening downwards) crosses the \(x\)-axis at \(x = -5\) and \(x = 1\). Thus, it has real roots.

Identify functions with no real roots (complex roots)

Quadratic functions that do not intersect or touch the \(x\)-axis have strictly complex (non-real) roots.

• Curve \(b\) (green, opening upwards) has its vertex at approximately \((2, 2)\) and lies entirely above the \(x\)-axis. It does not intersect the \(x\)-axis, meaning it has complex roots.
• Curve \(d\) (grey/dark grey, opening downwards) has its vertex at approximately \((1, -5)\) and lies entirely below the \(x\)-axis. It does not intersect the \(x\)-axis, meaning it has complex roots.

Determine the correct selections

The functions with complex roots are represented by the labels \(b\) and \(d\).

Answer:

• (a)
• (b) (Correct answer)
• (c)
• (d) (Correct answer)