Determine whether the equation defines y as a function of x. [tex]y=9 x^2-7 x+6[/tex] Does the equation define [tex]y[/tex] as a function of [tex]x[/tex]? A. Yes B. No
Answer (2)
The given equation is y = 9 x 2 − 7 x + 6 .
For every value of x , there is only one corresponding value of y .
The equation passes the vertical line test.
Therefore, the equation defines y as a function of x : Y es .
Explanation
Understanding the Problem We are given the equation y = 9 x 2 − 7 x + 6 and asked to determine if it defines y as a function of x . In simpler terms, we need to check if for every value of x , there is only one corresponding value of y .
Analyzing the Equation The given equation is a quadratic equation in the form y = a x 2 + b x + c , where a = 9 , b = − 7 , and c = 6 . For any real number x , we can substitute it into the equation and calculate a unique value for y . This is because the operations involved (squaring, multiplication, and addition) are well-defined and will always produce a single result.
Applying the Vertical Line Test To further illustrate, let's consider the vertical line test. If we were to graph this equation, it would be a parabola. A parabola opens upwards since the coefficient of x 2 is positive ( 0"> a = 9 > 0 ). The vertical line test states that if any vertical line intersects the graph at only one point, then the equation defines y as a function of x . Since a vertical line can only intersect a parabola at most twice, and in this case, each x yields a unique y , the vertical line test is satisfied.
Conclusion Therefore, the equation y = 9 x 2 − 7 x + 6 defines y as a function of x .
Examples
In real life, quadratic functions can model the trajectory of a ball thrown in the air. For each horizontal distance ( x ), there is only one corresponding height ( y ). This means the height of the ball is a function of its horizontal distance. Similarly, the relationship between the price of a product ( x ) and the quantity sold ( y ) can sometimes be modeled by a quadratic function, where each price corresponds to a unique quantity sold.
The equation y = 9 x 2 − 7 x + 6 defines y as a function of x because each value of x corresponds to one unique value of y . Since it passes the vertical line test as it forms a parabola, the answer is Y es .
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