Question

fill in the equation for this function.
y = ?(x - )^3 +

Explanation:

Step1: Recall the cubic function form

The general form of a cubic function with a horizontal shift \( h \) and vertical shift \( k \) is \( y = a(x - h)^3 + k \), where \( (h,k) \) is the point of inflection (the point where the curve changes concavity). From the graph, the point of inflection is at \( (0, -3) \), so \( h = 0 \) and \( k = -3 \). So the equation becomes \( y = a(x - 0)^3 - 3=a x^3 - 3 \).

Step2: Find the value of \( a \)

We can use another point on the graph. Let's take the point \( (1, -1) \) (since when \( x = 1 \), \( y=-1 \)). Substitute \( x = 1 \) and \( y=-1 \) into the equation \( y = a x^3 - 3 \):

$$ -1=a(1)^3 - 3 $$

Step3: Solve for \( a \)

Add 3 to both sides of the equation:

$$ -1 + 3=a(1) $$

$$ 2=a $$

Step4: Write the final equation

Substitute \( a = 2 \), \( h = 0 \), and \( k = -3 \) back into the general form \( y = a(x - h)^3 + k \). We get \( y = 2(x - 0)^3 - 3 \), which simplifies to \( y = 2(x - 0)^3 + (-3) \).

Answer:

The values are \( a = 2 \), \( h = 0 \), and \( k = -3 \). So the equation is \( y = \boldsymbol{2}(x - \boldsymbol{0})^3 + \boldsymbol{-3} \) (or \( y = 2(x - 0)^3 - 3 \)).