Use a graphing calculator to sketch the graph of the quadratic equation, and then state the domain and range. [tex]y=2 x^2-x+3[/tex]
A. D : all real numbers, R :[tex](y \geq 2.875)[/tex]
B. D : all real numbers, R: [tex](y \leq 3)[/tex]
C. D: [tex](y \leq-0.25)[/tex], R: [tex](y \leq 2.875)[/tex]
D. D : all real numbers, R :[tex](y \leq 3.125)[/tex]
Answer (2)
The domain of the quadratic function y = 2 x 2 − x + 3 is all real numbers, and its range is y ≥ 2.875 . The correct option is A.
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The domain of the quadratic function y = 2 x 2 − x + 3 is all real numbers.
Calculate the x-coordinate of the vertex: x v = 2 a − b = 4 1 = 0.25 .
Calculate the y-coordinate of the vertex: y v = 2 ( 0.25 ) 2 − 0.25 + 3 = 2.875 .
Since the parabola opens upwards, the range is y ≥ 2.875 . The answer is A .
Explanation
Understanding the Problem The problem asks us to find the domain and range of the quadratic equation y = 2 x 2 − x + 3 . We also need to select the correct option from the choices provided.
Determining the Domain The domain of a quadratic function is all real numbers because we can plug in any real number for x and get a real number for y .
Finding the Vertex - x-coordinate To find the range, we need to determine the vertex of the parabola. The x-coordinate of the vertex is given by the formula x v = 2 a − b , where a = 2 and b = − 1 in our equation. Thus, x v = 2 ( 2 ) − ( − 1 ) = 4 1 = 0.25.
Finding the Vertex - y-coordinate Now, we need to find the y-coordinate of the vertex by plugging x v = 0.25 into the equation: y v = 2 ( 0.25 ) 2 − 0.25 + 3 = 2 ( 16 1 ) − 4 1 + 3 = 8 1 − 8 2 + 8 24 = 8 23 = 2.875. So, the vertex of the parabola is at ( 0.25 , 2.875 ) .
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 represents the minimum value of the function. Therefore, the range of the function is all y values greater than or equal to the y-coordinate of the vertex, which is y ≥ 2.875 .
Selecting the Correct Option Comparing our findings with the given options, we see that option A matches our calculated domain and range:Domain: all real numbersRange: ( y ≥ 2.875 )
Examples
Quadratic equations are useful in many real-world scenarios, such as modeling the trajectory of a ball, designing parabolic mirrors, and optimizing business processes. For example, a company might use a quadratic equation to model the profit as a function of the price of a product and then find the price that maximizes the profit. The vertex of the parabola would represent the optimal price and the maximum profit.
