Question

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-axis from -10 to 10, y-axis from -10 to 10, two blue dots on x-axis, reset button, keypad, keyboard shortcuts)

Explanation:

Step1: Factor the polynomial

First, factor \( f(x) = x^2 - 3x - 4 \). We find two numbers that multiply to \(-4\) and add to \(-3\). Those numbers are \(-4\) and \(1\), so \( f(x)=(x - 4)(x+ 1) \). The roots are \( x=-1 \) and \( x = 4 \) (the blue dots on the x - axis).

Step2: Analyze the parabola's direction

The coefficient of \( x^2 \) is \( 1\) (positive), so the parabola opens upwards (like a "U" shape).

Step3: Determine positive/negative regions

• For \( x < - 1 \) (e.g., \( x=-2 \)): Substitute into \( f(x) \), \( f(-2)=(-2 - 4)(-2 + 1)=(-6)(-1)=6>0 \), so the graph is positive (above x - axis) here.
• For \( -1
• For \( x>4 \) (e.g., \( x = 5 \)): \( f(5)=(5 - 4)(5 + 1)=(1)(6)=6>0 \), so the graph is positive (above x - axis) here.

To sketch: Draw the parabola opening upwards with roots at \( x=-1 \) and \( x = 4 \). Plot the part above the x - axis for \( x < - 1 \) and \( x>4 \), and below the x - axis for \( -1

Answer:

The parabola \( f(x)=x^2 - 3x - 4 \) (opening upwards, roots at \( x=-1 \) and \( x = 4 \)) is positive for \( x < - 1 \) or \( x>4 \) (graph above x - axis) and negative for \( -14 \), below for \( -1