Which function is a quadratic function?
$y-3 x^2=3\left(x^2+5\right)+1$
$y^2-7 x=2\left(x^2+6\right)+7$
$y-2 x^2=6\left(x^3+5\right)-4$
$y-5 x=4(x+5)+9$
Answer (1)
Rearrange each equation to isolate y .
Simplify each equation and check if it matches the quadratic form y = a x 2 + b x + c , where a e q 0 .
Identify the equation that simplifies to a quadratic function.
The quadratic function is y = 6 x 2 + 16 .
Explanation
Understanding the Problem We are given four equations and asked to identify which one represents a quadratic function. A quadratic function has the general form y = a x 2 + b x + c , where a e q 0 . We need to rearrange each equation to isolate y and check if it matches the quadratic form.
Analyzing Each Equation Let's analyze each equation:
y − 3 x 2 = 3 ( x 2 + 5 ) + 1 Rearrange to isolate y : y = 3 x 2 + 3 ( x 2 + 5 ) + 1 . Simplify: y = 3 x 2 + 3 x 2 + 15 + 1 = 6 x 2 + 16 . This is a quadratic function.
y 2 − 7 x = 2 ( x 2 + 6 ) + 7 Rearrange to isolate y : y 2 = 2 ( x 2 + 6 ) + 7 + 7 x . Simplify: y 2 = 2 x 2 + 12 + 7 + 7 x = 2 x 2 + 7 x + 19 . Since y is squared, this is not a quadratic function.
y − 2 x 2 = 6 ( x 3 + 5 ) − 4 Rearrange to isolate y : y = 2 x 2 + 6 ( x 3 + 5 ) − 4 . Simplify: y = 2 x 2 + 6 x 3 + 30 − 4 = 6 x 3 + 2 x 2 + 26 . This is a cubic function.
y − 5 x = 4 ( x + 5 ) + 9 Rearrange to isolate y : y = 5 x + 4 ( x + 5 ) + 9 . Simplify: y = 5 x + 4 x + 20 + 9 = 9 x + 29 . This is a linear function.
Identifying the Quadratic Function From the analysis above, only the first equation, y − 3 x 2 = 3 ( x 2 + 5 ) + 1 , simplifies to a quadratic function: y = 6 x 2 + 16 .
Final Answer Therefore, the quadratic function is y = 6 x 2 + 16 .
Examples
Quadratic functions are incredibly useful in physics, engineering, and economics. For example, the trajectory of a projectile (like a ball thrown in the air) can be modeled using a quadratic function. Understanding quadratic functions helps us predict how high the ball will go and how far it will travel before landing. Similarly, engineers use quadratic functions to design bridges and arches, ensuring they can withstand specific loads. Economists use them to model cost and revenue curves to optimize business decisions.
