Use a graphing calculator to approximate the vertex of the graph of the parabola defined by the following equation:
[tex]$y=-2 x^2+7 x+4$[/tex]
a. [tex]$(1.75,-2.125)$[/tex]
b. [tex]$(1.75,4)$[/tex]
c. [tex]$(1.75,10.125)$[/tex]
d. [tex]$(-1.75,10.125)$[/tex]
Please select the best answer from the choices provided
Answer (2)
The vertex of the parabola defined by the equation y = − 2 x 2 + 7 x + 4 is calculated to be ( 1.75 , 10.125 ) . Therefore, the correct answer is (1.75, 10.125).
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Find the x-coordinate of the vertex using the formula x = − 2 a b , where a = − 2 and b = 7 . This gives x = 1.75 .
Substitute the x-coordinate into the equation y = − 2 x 2 + 7 x + 4 to find the y-coordinate.
Calculate y = − 2 ( 1.75 ) 2 + 7 ( 1.75 ) + 4 = 10.125 .
The vertex of the parabola is ( 1.75 , 10.125 ) , so the answer is ( 1.75 , 10.125 ) .
Explanation
Understanding the Problem We are given the equation of a parabola y = − 2 x 2 + 7 x + 4 and asked to find the vertex of its graph. The vertex of a parabola in the form y = a x 2 + b x + c is given by the point ( x , y ) , where x = − 2 a b and y is the value of the function at that x value.
Finding the x-coordinate First, we need to find the x-coordinate of the vertex. In our equation, a = − 2 and b = 7 . Using the formula x = − 2 a b , we have x = − 2 ( − 2 ) 7 = − − 4 7 = 4 7 = 1.75
Finding the y-coordinate Now, we need to find the y-coordinate of the vertex by substituting the x-coordinate, x = 1.75 , into the equation y = − 2 x 2 + 7 x + 4 :
y = − 2 ( 1.75 ) 2 + 7 ( 1.75 ) + 4 y = − 2 ( 3.0625 ) + 12.25 + 4 y = − 6.125 + 12.25 + 4 y = 6.125 + 4 y = 10.125
The Vertex Therefore, the vertex of the parabola is ( 1.75 , 10.125 ) .
Final Answer Comparing our result with the given options, we see that the correct answer is (1.75, 10.125).
Examples
Understanding the vertex of a parabola is useful in many real-world applications. For example, if you're launching a projectile, the vertex represents the maximum height the projectile will reach. Similarly, in business, if you're modeling profit as a function of price, the vertex can represent the price that maximizes your profit. Knowing how to find the vertex allows you to optimize various scenarios.
