Question

quiz 2.1: graphs of polynomial functions
score: 6/10 answered: 4/5
question 5
given the function $g(t) = (t - 6)(t + 7)(t - 1)$:
the coordinates of its $g$-intercept are
the coordinates of its $t$-intercepts are
check answer

Explanation:

For g - intercept:

Step1: Recall g - intercept definition

The g - intercept (y - intercept in terms of g(t)) occurs when \( t = 0 \). So we substitute \( t=0 \) into the function \( g(t)=(t - 6)(t + 7)(t - 1) \).

Step2: Substitute \( t = 0 \)

\( g(0)=(0 - 6)(0 + 7)(0 - 1)=(-6)\times(7)\times(-1) \)
First, calculate \( (-6)\times(7)=-42 \), then \( -42\times(-1) = 42 \).
So the g - intercept is at \( t = 0 \), \( g(0)=42 \), so the coordinates are \( (0,42) \).

For t - intercepts:

Step1: Recall t - intercept definition

The t - intercepts occur when \( g(t)=0 \). So we set \( (t - 6)(t + 7)(t - 1)=0 \).

Step2: Solve for t

Using the zero - product property, if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \). So:

• If \( t - 6=0 \), then \( t = 6 \).
• If \( t + 7=0 \), then \( t=-7 \).
• If \( t - 1=0 \), then \( t = 1 \).

So the t - intercepts are at \( t = 6 \), \( t=-7 \), and \( t = 1 \), so the coordinates are \( (6,0) \), \( (-7,0) \), and \( (1,0) \).

Answer:

• The coordinates of its g - intercept are \((0, 42)\).
• The coordinates of its t - intercepts are \((6, 0)\), \((-7, 0)\), \((1, 0)\)