Question

the graph of the function f(x) = 2x² - bx - 6 is shown. what is the value of b? explain.
the x-intercept(s) of the parabola is(are) (simplify your answer.)
the formula for this(these) point(s) can be used to determine that b = 4
options: (1, -8), (-1, -8), (0,6), (-3,0) and (1,0), (0, -6), (-1,0) and (3,0)

Explanation:

Step1: Recall x - intercept definition

The x - intercepts of a function \(y = f(x)\) are the values of \(x\) for which \(y=0\), i.e., \(f(x) = 0\). Given the function \(f(x)=2x^{2}-bx - 6\), and from the options, the x - intercepts are \((- 1,0)\) and \((3,0)\). So when \(x=-1\) or \(x = 3\), \(f(x)=0\).

Step2: Substitute x - intercept into function

Let's take one of the x - intercepts, say \(x = 3\). Substitute \(x = 3\) and \(f(x)=0\) into \(f(x)=2x^{2}-bx - 6\):
\[0=2\times(3)^{2}-b\times(3)-6\]
\[0 = 2\times9-3b - 6\]
\[0=18 - 3b-6\]
\[0 = 12-3b\]

Step3: Solve for b

Add \(3b\) to both sides of the equation \(0 = 12-3b\):
\[3b=12\]
Divide both sides by 3:
\[b = 4\]
We can also check with \(x=-1\). Substitute \(x=-1\) and \(f(x) = 0\) into \(f(x)=2x^{2}-bx - 6\):
\[0=2\times(-1)^{2}-b\times(-1)-6\]
\[0=2 + b-6\]
\[0=b - 4\]
\[b = 4\]

Answer:

The x - intercept(s) of the parabola is(are) \((-1,0)\) and \((3,0)\). The value of \(b = 4\)