Which statements about the graph of the function [tex]$f(x) = 2x^2 - x - 6$[/tex] are true? Select two options.
The domain of the function is [tex]$\left\{x \left\lvert\, x \geq \frac{1}{4}\right.\right\}$[/tex].
The range of the function is all real numbers.
The vertex of the function is [tex]$\left(\frac{1}{4},-6 \frac{1}{8}\right)$[/tex].
The function has two [tex]$x$[/tex]-intercepts.
The function is increasing over the interval [tex]$\left(-6 \frac{1}{8}, \infty \right)$[/tex].
Answer (1)
The vertex of the function is ( 4 1 , − 6 8 1 ) .
The function has two x -Intercepts: x = − 1.5 and x = 2 .
The domain of the function is all real numbers.
The function is increasing over the interval ( 4 1 , ∞ ) .
The two true statements are the vertex and the x-intercepts. \boxed{The vertex of the function is \left(\frac{1}{4},-6 \frac{1}{8}\right) an d t h e f u n c t i o nha s tw o x -Intercepts.}
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 The domain of a quadratic function is all real numbers, since we can plug in any real number for x and get a real number out. Therefore, the statement 'The domain of the function is { x x ≥ 4 1 } ' is false.
Finding the Vertex The vertex of the parabola is given by the formula x = − 2 a b , where a = 2 and b = − 1 . Thus, the x-coordinate of the vertex is x = − 2 ( 2 ) − 1 = 4 1 . To find the y-coordinate, we plug this value into the function: f ( 4 1 ) = 2 ( 4 1 ) 2 − 4 1 − 6 = 2 ( 16 1 ) − 4 1 − 6 = 8 1 − 8 2 − 8 48 = − 8 49 = − 6 8 1 . Therefore, the vertex is ( 4 1 , − 6 8 1 ) . So, the statement 'The vertex of the function is ( 4 1 , − 6 8 1 ) ' is true.
Determining the Range Since the coefficient of the x 2 term is positive ( 0"> a = 2 > 0 ), the parabola opens upwards. This means the vertex is the minimum point of the function. Therefore, the range of the function is all real numbers greater than or equal to the y-coordinate of the vertex, which is − 6 8 1 . So the range is { y y ≥ − 6 8 1 } . Thus, the statement 'The range of the function is all real numbers' is false.
Finding the x-intercepts To find the x-intercepts, we set f ( x ) = 0 and solve for x : 2 x 2 − x − 6 = 0 . We can factor this quadratic as ( 2 x + 3 ) ( x − 2 ) = 0 . This gives us two solutions: 2 x + 3 = 0 ⇒ x = − 2 3 = − 1.5 and x − 2 = 0 ⇒ x = 2 . Since there are two distinct real solutions, the function has two x-intercepts. Thus, the statement 'The function has two x -Intercepts' is true.
Finding the Increasing Interval Since the parabola opens upwards, the function is increasing to the right of the vertex. The x-coordinate of the vertex is 4 1 . Therefore, the function is increasing over the interval ( 4 1 , ∞ ) . The statement 'The function is increasing over the interval ( − 6 8 1 , ∞ ) ' is false, since − 6 8 1 is the y-coordinate of the vertex, not the x-coordinate.
Final Answer The two true statements are:
The vertex of the function is ( 4 1 , − 6 8 1 ) .
The function has two x -Intercepts.
Examples
Understanding the properties of quadratic functions, such as finding the vertex and x-intercepts, is crucial in various real-world applications. For instance, engineers use these concepts to design parabolic reflectors for satellite dishes or solar ovens, optimizing the focus of incoming signals or sunlight. Similarly, economists model cost and revenue curves using quadratic functions to determine break-even points and maximize profits. The vertex represents the point of maximum profit or minimum cost, while the x-intercepts indicate the points where revenue equals cost.
