Use a graphing calculator to sketch the graph of the quadratic equation, and then state the domain and range: [tex]y=-5 x^2+3 x+10[/tex]
a. [tex]D :(x \geq 0.3)[/tex]
c. D: [tex](x \geq 0.3)[/tex]
R : all real numbers
R: [tex](y \leq 10)[/tex]
b. D: all real numbers
d. D: all real numbers
R: [tex](y \geq 10.45)[/tex]
R: [tex](y \leq 10.45)[/tex]
Please select the best answer from the choices provided
A
B
C
D
Answer (2)
The domain of the quadratic equation y = − 5 x 2 + 3 x + 10 is all real numbers, and the range is y ≤ 10.45 . Therefore, the best answer according to the provided choices is option D. The vertex, which establishes the maximum value of the function, is found at the coordinate (0.3, 10.45).
;
The domain of the quadratic function y = − 5 x 2 + 3 x + 10 is all real numbers.
Calculate the x-coordinate of the vertex: x = − b / ( 2 a ) = − 3/ ( 2 ∗ ( − 5 )) = 0.3 .
Calculate the y-coordinate of the vertex: y = − 5 ( 0.3 ) 2 + 3 ( 0.3 ) + 10 = 10.45 .
Since the parabola opens downwards, the range is y ≤ 10.45 . Therefore, the answer is D .
Explanation
Problem Analysis We are given the quadratic equation y = − 5 x 2 + 3 x + 10 . Our goal is to determine its domain and range and choose the correct option from the provided choices.
Determining the Domain The domain of a quadratic function is all real numbers because it is a polynomial. There are no restrictions on the values of x that can be input into the function.
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 = − b / ( 2 a ) , where a = − 5 and b = 3 . Thus, x = − 3/ ( 2 ∗ − 5 ) = 3/10 = 0.3 .
Finding the Vertex - y-coordinate Now, we find the y-coordinate of the vertex by substituting x = 0.3 into the equation: y = − 5 ( 0.3 ) 2 + 3 ( 0.3 ) + 10 = − 5 ( 0.09 ) + 0.9 + 10 = − 0.45 + 0.9 + 10 = 10.45 .
Determining the Range Since the coefficient of the x 2 term is negative ( a = − 5 ), the parabola opens downwards. This means the vertex represents the maximum point of the parabola. Therefore, the range is all y values less than or equal to the y-coordinate of the vertex, which is 10.45 . So, the range is y ≤ 10.45 .
Selecting the Correct Option Comparing our findings with the given options, we see that option d matches our results: Domain: all real numbers, Range: ( y ≤ 10.45 ) .
Examples
Imagine you're designing a suspension bridge, and the cable's curve is modeled by a quadratic equation. Understanding the domain (possible distances between supports) and range (maximum height of the cable) is crucial for ensuring the bridge's safety and stability. Similarly, in projectile motion, the path of a ball thrown in the air can be described by a quadratic equation. The domain represents the time the ball is in the air, and the range represents the maximum height the ball reaches. These concepts are also applicable in business, where quadratic functions can model profit curves, helping to determine optimal production levels.
