Question

question 1 of 10
which of the following graphs is described by the function given below?
y = 3x² + 7x + 2
four graphs are shown here
click here for long description

a. graph a
b. graph b
c. graph c
d. graph d

Explanation:

Step1: Analyze the quadratic function

The function is \( y = 3x^2 + 7x + 2 \), which is a quadratic function in the form \( y = ax^2+bx + c \) with \( a = 3 \), \( b = 7 \), \( c = 2 \). Since \( a=3>0 \), the parabola opens upwards.

Step2: Find the x - intercepts

To find the x - intercepts, set \( y = 0 \), so we solve the equation \( 3x^2+7x + 2=0 \).
We can factor the quadratic: \( 3x^2+7x + 2=(3x + 1)(x+2)=0 \).
Setting each factor equal to zero:

• For \( 3x+1 = 0 \), we get \( x=-\frac{1}{3}\approx - 0.33\)
• For \( x + 2=0 \), we get \( x=-2\)

So the x - intercepts are at \( x=-2 \) and \( x =-\frac{1}{3}\).

Step3: Analyze the vertex (optional for elimination)

The x - coordinate of the vertex of a parabola \( y=ax^2+bx + c \) is given by \( x=-\frac{b}{2a} \). Here, \( x =-\frac{7}{2\times3}=-\frac{7}{6}\approx - 1.17\)

Now, let's analyze the graphs:

• Graph A: Likely does not have x - intercept at \( x=-2 \) (from visual inspection of typical options, since the x - intercepts we found are \( x=-2 \) and \( x\approx - 0.33\))
• Graph B: Since the parabola opens upwards and has x - intercepts at \( x=-2 \) and \( x\approx - 0.33\), this graph should match.
• Graph C: Unlikely to have x - intercept at \( x=-2 \) (based on the x - intercepts we calculated)
• Graph D: Unlikely to have x - intercept at \( x=-2 \) (based on the x - intercepts we calculated)

Answer:

B. Graph B