Question

1. write the equation and draw the graph of the function if the parent cubic function is compressed vertically by a factor of 1/2, reflected over the x - axis and then translated 2 units left and 3 units down.

equation

1. a cubic function with general equation f(x) = a(x - h)³ + k passes through the points (-3, -2) and (-4, 1), where (-4, 1) is the inflection point. write the equation of the function.

Explanation:

Step1: Identify the inflection point

The inflection point of the cubic function \( f(x) = a(x - h)^3 + k \) is at \( (h, k) \). Given that the inflection point is \( (-4, 1) \), we have \( h = -4 \) and \( k = 1 \). So the function becomes \( f(x) = a(x + 4)^3 + 1 \).

Step2: Substitute the given point

The function passes through the point \( (-3, -2) \). Substitute \( x = -3 \) and \( f(x) = -2 \) into the equation:

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

Simplify the equation:

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

$$ -2 = a + 1 $$

Subtract 1 from both sides:

$$ a = -2 - 1 = -3 $$

Step3: Write the final equation

Substitute \( a = -3 \), \( h = -4 \), and \( k = 1 \) back into the general form:

$$ f(x) = -3(x + 4)^3 + 1 $$

Answer:

\( f(x) = -3(x + 4)^3 + 1 \)