Question

find the vertex of the graph of the quadratic function. determine whether the graph opens upward or downward, find any intercepts, and sketch the graph. f(x)=2x² + 8x + 14 the vertex is (simplify your answer. type an ordered pair.) (-2,6). does the graph open upward or downward? a. the parabola opens upward. b. the parabola opens downward. find any x-intercepts of the graph. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the x-intercept(s) is(are) (simplify your answer. type an ordered pair. use a comma to separate answers as needed.) b. there is no x-intercept.

Explanation:

Part 1: Direction of the Parabola

Step1: Recall the form of a quadratic function

The general form of a quadratic function is \( f(x) = ax^2 + bx + c \). If \( a>0 \), the parabola opens upward; if \( a < 0 \), it opens downward.

Step2: Identify the value of \( a \) in the given function

For the function \( f(x)=2x^{2}+8x + 14 \), the coefficient of \( x^{2} \) (i.e., \( a \)) is \( 2 \). Since \( 2>0 \), the parabola opens upward.

Part 2: Finding x - intercepts

Step1: Set \( f(x)=0 \) to find x - intercepts

To find the x - intercepts, we set \( f(x)=0 \), so we have the equation \( 2x^{2}+8x + 14=0 \).

Step2: Simplify the equation

Divide the entire equation by \( 2 \): \( x^{2}+4x + 7 = 0 \).

Step3: Use the discriminant formula

The discriminant of a quadratic equation \( ax^{2}+bx + c = 0 \) is given by \( D=b^{2}-4ac \). For the equation \( x^{2}+4x + 7 = 0 \), \( a = 1 \), \( b = 4 \), and \( c = 7 \).
Calculate the discriminant: \( D=(4)^{2}-4\times1\times7=16 - 28=- 12 \).

Step4: Analyze the discriminant

Since the discriminant \( D=-12<0 \), the quadratic equation \( x^{2}+4x + 7 = 0 \) has no real solutions. This means the graph of the function \( f(x)=2x^{2}+8x + 14 \) has no x - intercepts.

Answer:

for the direction of the parabola:
A. The parabola opens upward.