Which statements about the graph of the function [tex]f(x)= 2 x^2-x-6[/tex] are true? Select two options.
A. The domain of the function is [tex] \left\{x \left\lvert\, x \geq \frac{1}{4}\right.\right\}[/tex].
B. The range of the function is all real numbers.
C. The vertex of the function is [tex]\left(\frac{1}{4},-6 \frac{1}{8}\right)[/tex].
D. The function has two x-intercepts.
E. The function is increasing over the interval [tex]\left(-6 \frac{1}{8}, \infty\right)[/tex].
Answer (2)
Determine that the domain of the quadratic function is all real numbers.
Calculate the vertex of the function using x = − b / ( 2 a ) and f ( x ) .
Find the x-intercepts by solving the quadratic equation 2 x 2 − x − 6 = 0 using the quadratic formula.
Conclude that the vertex is ( 4 1 , − 6 8 1 ) and there are two x-intercepts, making statements 3 and 4 true. St a t e m e n t s 3 an d 4
Explanation
Analyzing the Problem We are given the quadratic function f ( x ) = 2 x 2 − x − 6 and asked to determine which two of the given statements about its graph are true. Let's analyze each statement.
Checking the Domain Statement 1: The domain of the function is { x x ≥ 4 1 } .
Since f ( x ) is a quadratic function, its domain is all real numbers. Therefore, this statement is false.
Checking the Range Statement 2: The range of the function is all real numbers. Since the coefficient of the x 2 term is positive (2 > 0), the parabola opens upwards. This means the function has a minimum value at its vertex, and the range will be all real numbers greater than or equal to the y-coordinate of the vertex. Therefore, this statement is false.
Finding the Vertex Statement 3: The vertex of the function is ( 4 1 , − 6 8 1 ) .
The x-coordinate of the vertex is given by x = − b / ( 2 a ) , where a = 2 and b = − 1 . So, x = − ( − 1 ) / ( 2 ∗ 2 ) = 1/4 .
The y-coordinate of the vertex is f ( 1/4 ) = 2 ∗ ( 1/4 ) 2 − ( 1/4 ) − 6 = 2 ∗ ( 1/16 ) − 1/4 − 6 = 1/8 − 2/8 − 48/8 = − 49/8 = − 6 8 1 .
So the vertex is indeed ( 4 1 , − 6 8 1 ) . Therefore, this statement is true.
Finding the X-Intercepts Statement 4: The function has two x-intercepts. To find the x-intercepts, we set f ( x ) = 0 and solve for x: 2 x 2 − x − 6 = 0 .
We can use the quadratic formula: x = 2 a − b ± b 2 − 4 a c = 2 ( 2 ) − ( − 1 ) ± ( − 1 ) 2 − 4 ( 2 ) ( − 6 ) = 4 1 ± 1 + 48 = 4 1 ± 49 = 4 1 ± 7 .
So x = 4 1 + 7 = 2 and x = 4 1 − 7 = − 2 3 .
Since we have two distinct real roots, the function has two x-intercepts. Therefore, this statement is true.
Finding the Increasing Interval Statement 5: The function is increasing over the interval ( − 6 8 1 , ∞ ) .
Since the parabola opens upwards, the function is increasing to the right of the vertex. The x-coordinate of the vertex is 4 1 , so the function is increasing over the interval ( 4 1 , ∞ ) . Therefore, this statement is false.
Conclusion The true statements are: Statement 3: The vertex of the function is ( 4 1 , − 6 8 1 ) .
Statement 4: The function has two x-intercepts.
Examples
Understanding the properties of quadratic functions, like finding the vertex and x-intercepts, is crucial in many real-world applications. For example, engineers use this knowledge to design parabolic reflectors for satellite dishes or solar ovens, optimizing the focus of signals or heat. Similarly, in business, understanding the vertex of a cost function can help determine the production level that minimizes costs. These applications highlight how a solid grasp of quadratic functions translates into practical problem-solving skills across various fields.
The true statements about the function f ( x ) = 2 x 2 − x − 6 are that the vertex is ( 4 1 , − 6 8 1 ) and the function has two x-intercepts. Therefore, options C and D are correct.
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