What are the coordinates of the turning point of the graph of the quadratic equation
$y=2 x^2+5 x-2$
A. $\left(-\frac{5}{4},-\frac{41}{16}\right)$
B. $\left(-\frac{5}{4},-\frac{41}{8}\right)$
C. $\left(\frac{5}{4},-\frac{41}{16}\right)$
D. $\left(-\frac{5}{2},-\frac{41}{8}\right)$
E. $\quad\left(-\frac{5}{2},-\frac{41}{16}\right)$
Answer (2)
Find the x-coordinate of the turning point using the formula x = − 2 a b , which gives x = − 4 5 .
Substitute the x-coordinate into the original equation to find the y-coordinate: y = 2 ( − 4 5 ) 2 + 5 ( − 4 5 ) − 2 .
Simplify the expression to find the y-coordinate: y = − 8 41 .
The coordinates of the turning point are ( − 4 5 , − 8 41 ) .
Explanation
Understanding the Problem We are given the quadratic equation y = 2 x 2 + 5 x − 2 and asked to find the coordinates of its turning point. The turning point of a quadratic equation is its vertex, which represents either the minimum or maximum value of the function.
Finding the x-coordinate For a quadratic equation in the form y = a x 2 + b x + c , the x-coordinate of the vertex is given by the formula x = − 2 a b . In our equation, a = 2 and b = 5 .
Calculating the x-coordinate Substituting the values of a and b into the formula, we get x = − 2 ( 2 ) 5 = − 4 5 .
Finding the y-coordinate Now that we have the x-coordinate, we can find the y-coordinate by substituting x = − 4 5 back into the original equation: y = 2 ( − 4 5 ) 2 + 5 ( − 4 5 ) − 2 .
Calculating the y-coordinate Let's simplify this expression:
First, we square − 4 5 to get 16 25 . Then, we multiply by 2: 2 ( 16 25 ) = 16 50 = 8 25 .
Next, we multiply 5 ( − 4 5 ) = − 4 25 .
So, we have y = 8 25 − 4 25 − 2 . To combine these terms, we need a common denominator, which is 8. Thus, we rewrite the equation as y = 8 25 − 8 50 − 8 16 .
Combining the fractions, we get y = 8 25 − 50 − 16 = 8 − 41 .
Final Answer Therefore, the coordinates of the turning point are ( − 4 5 , − 8 41 ) .
Examples
Understanding the turning point of a quadratic equation is useful in many real-world applications. For example, if you are designing a suspension bridge, the equation describing the curve of the bridge can be modeled as a quadratic. Finding the turning point helps determine the lowest point of the bridge, which is crucial for ensuring sufficient clearance and structural integrity. Similarly, in business, if you model profit as a quadratic function of production quantity, the turning point indicates the production level that maximizes profit.
The turning point of the quadratic equation y = 2 x 2 + 5 x − 2 is found to be at the coordinates ( − 4 5 , − 8 41 ) . Therefore, the correct answer is option B. This involves calculating the x-coordinate using x = − 2 a b and substituting back to find the y-coordinate.
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