Question

fill in the equation for this function.

y = ?(x - )^3 +

Explanation:

Step1: Identify the vertex form of cubic function

The general form of a cubic function in vertex form is \( y = a(x - h)^3 + k \), where \((h, k)\) is the vertex (the point where the function changes direction). From the graph, the vertex (the black dot at the "corner" of the curve) is at \((3, 3)\), so \( h = 3 \) and \( k = 3 \). So now we have \( y = a(x - 3)^3 + 3 \).

Step2: Find the value of \( a \)

We can use another point on the graph to find \( a \). Let's take the point \((4, 1)\) (the other black dot). Substitute \( x = 4 \) and \( y = 1 \) into the equation:
\( 1 = a(4 - 3)^3 + 3 \)
Simplify inside the parentheses: \( 4 - 3 = 1 \), so the equation becomes \( 1 = a(1)^3 + 3 \)
Which simplifies to \( 1 = a + 3 \)
Subtract 3 from both sides: \( a = 1 - 3 = -2 \)

Answer:

\( y = -2(x - 3)^3 + 3 \)