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 and y axes, reset button, and a parabola drawn partially; x-axis from -10 to 10, y-axis from -10 to 10, two blue dots on the x-axis, a reset button on the left, and a keypad/keyboard shortcuts section on the right)

Explanation:

Step1: Find roots of the polynomial

First, factor the quadratic function \( f(x) = x^2 - 3x - 4 \). We factor it as \( f(x)=(x - 4)(x+ 1) \). Setting \( f(x) = 0 \), we get the roots \( x=- 1 \) and \( x = 4 \) (wait, but in the graph, the roots seem to be at \( x = 0 \)? Wait, no, maybe I miscalculated. Wait, \( x^2-3x - 4=(x - 4)(x + 1) \), so roots at \( x=-1 \) and \( x = 4 \). But the graph in the image has roots at \( x = 0 \)? Wait, maybe the graph is a typo, but let's proceed with the function. The quadratic \( y=x^2-3x - 4 \) is a parabola opening upwards (since the coefficient of \( x^2 \) is positive). So the parabola is below the x - axis (negative) between the roots \( x=-1 \) and \( x = 4 \), and above the x - axis (positive) when \( x < - 1 \) or \( x>4 \).

Step2: Analyze the graph's positivity/negativity

For the given function \( f(x)=x^{2}-3x - 4 \), since the leading coefficient is positive, the parabola opens upwards. The roots are found by solving \( x^{2}-3x - 4=0 \). Using the quadratic formula \( x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \) for \( ax^{2}+bx + c = 0 \), here \( a = 1 \), \( b=-3 \), \( c=-4 \). So \( x=\frac{3\pm\sqrt{9 + 16}}{2}=\frac{3\pm5}{2} \). Thus, \( x=\frac{3 + 5}{2}=4 \) and \( x=\frac{3-5}{2}=-1 \). So the parabola is negative (below the x - axis) in the interval \( (-1,4) \) and positive (above the x - axis) in \( (-\infty,-1)\cup(4,\infty) \). To sketch the graph, we plot the positive regions (above x - axis) outside the interval between the roots and the negative region (below x - axis) between the roots.

Answer:

To sketch the graph: The parabola \( f(x)=x^{2}-3x - 4 \) (opening upwards) is positive (above the x - axis) for \( x < - 1 \) and \( x>4 \), and negative (below the x - axis) for \( -1 < x < 4 \). So in the graph, the regions above the x - axis (positive) are to the left of \( x=-1 \) and to the right of \( x = 4 \), and the region below the x - axis (negative) is between \( x=-1 \) and \( x = 4 \). (Note: The given graph in the problem might have a typo in the root positions, but the analysis is based on the function \( f(x)=x^{2}-3x - 4 \).)