Use a graphing calculator to sketch the graph of the quadratic equation, and then state the domain and range.
[tex]y=-0.5 x^2+0.7 x-5.1[/tex]
a. D: all real numbers
R: [tex](y \leq-5.1)[/tex]
b. D: all real numbers
R: [tex](y \geq 4.855)[/tex]
c. D: all real numbers
R: [tex](y \leq-4.855)[/tex]
d. D: all real numbers
R: [tex](y \leq-5.345)[/tex]
Please select the best answer from the choices provided
Answer (2)
The domain of the quadratic equation is all real numbers, and the range is y ≤ − 4.855 . The correct answer is option (C).
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The domain of the quadratic equation is all real numbers.
Calculate the x-coordinate of the vertex using x = 2 a − b , which gives x = 0.7 .
Substitute x = 0.7 into the equation to find the y-coordinate of the vertex, resulting in y = − 4.855 .
Since the parabola opens downwards, the range is y ≤ − 4.855 . The final answer is C.
Explanation
Understanding the Problem We are given the quadratic equation y = − 0.5 x 2 + 0.7 x − 5.1 . Our goal is to determine its domain and range and select the correct option from the choices provided.
Determining the Domain The domain of a quadratic equation is all real numbers because we can input any real number for x and obtain a real number for y .
Finding the Vertex To find the range, we need to determine the vertex of the parabola. Since the coefficient of the x 2 term is negative (-0.5), the parabola opens downwards, meaning it has a maximum point. The vertex's x-coordinate is given by the formula x = 2 a − b , where a = − 0.5 and b = 0.7 .
Calculating the x-coordinate of the Vertex Let's calculate the x-coordinate of the vertex: x = 2 ( − 0.5 ) − 0.7 = − 1 − 0.7 = 0.7
Calculating the y-coordinate of the Vertex Now, we substitute x = 0.7 into the quadratic equation to find the y-coordinate of the vertex: y = − 0.5 ( 0.7 ) 2 + 0.7 ( 0.7 ) − 5.1
Determining the Vertex Coordinates Let's calculate the y-coordinate:
y = − 0.5 ( 0.49 ) + 0.49 − 5.1 = − 0.245 + 0.49 − 5.1 = 0.245 − 5.1 = − 4.855
So, the vertex is at ( 0.7 , − 4.855 ) .
Determining the Range Since the parabola opens downwards, the range is all y values less than or equal to the y-coordinate of the vertex. Therefore, the range is y ≤ − 4.855 .
Final Answer The domain is all real numbers, and the range is y ≤ − 4.855 . This corresponds to option C.
Examples
Quadratic equations are useful in modeling projectile motion, such as the path of a ball thrown in the air. The domain represents the possible horizontal distances the ball can travel, while the range represents the possible heights the ball can reach. By finding the vertex, we can determine the maximum height the ball will reach. This has applications in sports, engineering, and physics.
