Question

fill in the equation for this function.
y = ?(x - )^4 +

Explanation:

Step1: Identify the vertex form of a quartic function

The general form of a quartic function in vertex form is \( y = a(x - h)^4 + k \), where \((h, k)\) is the vertex of the graph. From the graph, we can see that the vertex is at \((-3, -1)\). So, \( h = -3 \) and \( k = -1 \). Substituting these into the equation, we get \( y = a(x - (-3))^4 + (-1) \), which simplifies to \( y = a(x + 3)^4 - 1 \).

Step2: Find the value of \( a \)

We can use the other point on the graph to find \( a \). The graph passes through the point \((-2, 1)\). Substitute \( x = -2 \) and \( y = 1 \) into the equation \( y = a(x + 3)^4 - 1 \):

$$ 1 = a(-2 + 3)^4 - 1 $$

Simplify the expression inside the parentheses: \( -2 + 3 = 1 \). Then we have:

$$ 1 = a(1)^4 - 1 $$

Since \( 1^4 = 1 \), this becomes:

$$ 1 = a(1) - 1 $$

Add 1 to both sides:

$$ 1 + 1 = a $$

So, \( a = 2 \).

Step3: Write the final equation

Substitute \( a = 2 \), \( h = -3 \), and \( k = -1 \) back into the vertex form equation. We get \( y = 2(x + 3)^4 - 1 \), which matches the form \( y = [?](x - [\square])^4 + [\square] \) (note that \( x - (-3) = x + 3 \), so the middle box is \(-3\) in the form \( x - h \) where \( h = -3 \)).

Answer:

The equation is \( y = 2(x - (-3))^4 + (-1) \), so the values are \( 2 \), \( -3 \), and \( -1 \). Filling in the boxes, we have:

• The first box (coefficient \( a \)): \( 2 \)
• The second box (value of \( h \)): \( -3 \)
• The third box (value of \( k \)): \( -1 \)

So the equation is \( y = \boldsymbol{2}(x - \boldsymbol{-3})^4 + \boldsymbol{-1} \) (or written as \( y = 2(x + 3)^4 - 1 \)).