QuestionWrite an equation for the polynomial graphed below $y(x)=$ $\square$

Studdy Solution

STEP 1

1. The polynomial has roots at $x = -4, -1, 2,$ and $3$ .
2. The polynomial is likely to be of the fourth degree, as it has four roots.
3. The leading coefficient affects the vertical stretch/compression and the orientation (upwards/downwards) of the graph.
4. The polynomial can be expressed in the form $y(x) = a(x + 4)(x + 1)(x - 2)(x - 3)$ , where $a$ is the leading coefficient.

STEP 2

1. Write the polynomial expression using the identified roots.
2. Determine the leading coefficient $a$ using additional information from the graph (e.g., a specific point).
3. Verify the polynomial equation against the graph.

STEP 3

Write the polynomial expression using the identified roots.
$y(x) = a(x + 4)(x + 1)(x - 2)(x - 3)$

STEP 4

Identify a point on the graph to determine the leading coefficient $a$ . For example, if the graph passes through the point $(0, y_0)$ .
Let's assume the graph passes through $(0, c)$ .

STEP 5

Substitute $x = 0$ into the polynomial and solve for $a$ using the known value $y(0) = c$ .
$y(0) = a(0 + 4)(0 + 1)(0 - 2)(0 - 3) = c$ $a \cdot 4 \cdot 1 \cdot (-2) \cdot (-3) = c$ $a \cdot 24 = c$ $a = \frac{c}{24}$

STEP 6

Substitute the value of $a$ back into the polynomial equation.
$y(x) = \frac{c}{24} (x + 4)(x + 1)(x - 2)(x - 3)$

STEP 7

Verify the polynomial equation by comparing its graph to the given graph. Adjust $c$ if necessary based on the y-intercept or other points.

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