Which statements about the graph of the function [tex]f(x)= 2 x^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 found using x v = − 2 a b and f ( x v ) , resulting in ( 4 1 , − 6 8 1 ) .
The x-intercepts are found by solving 2 x 2 − x − 6 = 0 , which factors to ( 2 x + 3 ) ( x − 2 ) = 0 , giving x = − 2 3 and x = 2 .
The parabola opens upwards, so the range is [ − 6 8 1 , ∞ ) and the function increases over ( 4 1 , ∞ ) .
Therefore, the two true statements are that the vertex is ( 4 1 , − 6 8 1 ) and the function has two x-intercepts. \boxed{The vertex of the function is (\frac{1}{4},-6 \frac{1}{8}) an d t h e f u n c t i o nha s tw o x -intercepts.}
Explanation
Understanding 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. The statements concern the domain, range, vertex, x-intercepts, and increasing interval of the function.
Determining the Domain The domain of a quadratic function is all real numbers since we can plug in any real number for x . Therefore, the statement "The domain of the function is { x x ≥ 4 1 } " is false.
Finding the Vertex To find the vertex of the parabola, we use the formula x v = − 2 a b , where a = 2 and b = − 1 . Thus, x v = − 2 ( 2 ) − 1 = 4 1 . To find the y-coordinate of the vertex, we evaluate 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.
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 . Thus, the x-intercepts are x = − 2 3 = − 1.5 and x = 2 . Since there are two distinct x-intercepts, the statement "The function has two x -intercepts" 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 that the vertex is the minimum point of the function. Therefore, the range of the function is [ − 6 8 1 , ∞ ) . Thus, the statement "The range of the function is all real numbers" is false.
Determining the Increasing Interval Since the parabola opens upwards, the function is increasing for x_v"> x > x v , where x v is the x-coordinate of the vertex. Since x v = 4 1 , the function is increasing over the interval ( 4 1 , ∞ ) . Thus, the statement "The function is increasing over the interval ( − 6 8 1 , ∞ ) " is false.
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 quadratic functions is crucial in various real-world applications. For instance, engineers use quadratic equations to model the trajectory of projectiles, such as rockets or balls. By analyzing the vertex, x-intercepts, and increasing/decreasing intervals, they can determine the maximum height, range, and optimal launch angle for these projectiles. Similarly, economists use quadratic functions to model cost and revenue curves, helping businesses optimize production and pricing strategies to maximize profits. The vertex in this context represents the point of maximum profit or minimum cost.
