Question

question 15 of 24 step 2 of 2
consider the following polynomial function.
$f(x) = x^2 - 3x - 4$
step 2 of 2: sketch the graph by identifying where the graph is positive and where
the graph is negative.
answer 3 points
select the area above the axis to plot the positive curve for that region, and select the
area below the axis to plot the negative curve for that region. use the reset button to
remove all curves.
(graph with x and y axes, grid, reset button, keypad, keyboard shortcuts)

Explanation:

Step1: Find roots of the polynomial

To find where the graph is positive or negative, first find the roots of \( f(x)=x^{2}-3x - 4 \). Factor the quadratic: \( x^{2}-3x - 4=(x - 4)(x+ 1) \). Set \( f(x) = 0 \), so \( x - 4 = 0 \) or \( x+1 = 0 \), giving roots \( x=-1 \) and \( x = 4 \).

Step2: Analyze the parabola's direction

The coefficient of \( x^{2} \) is positive (\( a = 1>0 \)), so the parabola opens upwards.

Step3: Determine positive/negative regions

For a parabola opening upwards, the graph is negative between the roots (\( -14 \)). So, plot the positive curve (above x - axis) for \( x < - 1 \) and \( x>4 \), and the negative curve (below x - axis) for \( -1

Answer:

• Positive regions: \( x < - 1 \) and \( x>4 \) (plot above x - axis in these intervals)
• Negative region: \( -1