Question

topic 3 characteristics of polynomial functions skills practice continued 3 ( c(x) = x^3 ) ( g(x) = \frac{1}{4}c(x) ) reference points on ( c(x) ): (0, 0), (1, 1), (2, 16) corresponding points on ( g(x) ): (table with arrows) (graph of ( c(x) ) and ( g(x) ) with points (-2, -8), (-1, -1), (0, 0), (1, 1), (2, 8)) topic 3 characteristics of polynomial functions skills practice continued 4 ( m(x) = x^4 ) ( p(x) = m(x - 1) ) reference points on ( m(x) ): (0, 0), (1, 1), (2, 16) corresponding points on ( p(x) ): (table with arrows) (graph of ( m(x) ) and ( p(x) ) with points (-2, 16), (-1, 1), (0, 0), (1, 1), (2, 16))

Explanation:

Step1: Analyze the transformation for \( p(x) = m(x - 1) \)

The function \( p(x) \) is a horizontal shift of \( m(x) \). For a function \( y = f(x - h) \), it is a shift of \( f(x) \) by \( h \) units to the right. Here, \( h = 1 \), so we shift each point on \( m(x) \) 1 unit to the right.

Step2: Find the corresponding point for \( (0, 0) \) on \( m(x) \)

For the point \( (0, 0) \) on \( m(x) \), shifting 1 unit right (add 1 to the x - coordinate, y - coordinate remains the same) gives \( (0 + 1, 0)=(1, 0) \).

Step3: Find the corresponding point for \( (1, 1) \) on \( m(x) \)

For the point \( (1, 1) \) on \( m(x) \), shifting 1 unit right gives \( (1 + 1, 1)=(2, 1) \).

Step4: Find the corresponding point for \( (2, 16) \) on \( m(x) \)

For the point \( (2, 16) \) on \( m(x) \), shifting 1 unit right gives \( (2 + 1, 16)=(3, 16) \).

Answer:

For the point \((0,0)\) on \(m(x)\), the corresponding point on \(p(x)\) is \((1,0)\); for \((1,1)\) on \(m(x)\), the corresponding point on \(p(x)\) is \((2,1)\); for \((2,16)\) on \(m(x)\), the corresponding point on \(p(x)\) is \((3,16)\)

(If we assume the table needs to be filled:

Reference Points on \(m(x)\)	Corresponding Points on \(p(x)\)
\((1,1)\)	\((2,1)\)
\((2,16)\)	\((3,16)\)	)