Writing a quadratic function that represents a parabola that touches but does not cross the $x$-axis at $x=-6$. Which function could Heather be writing?
A. $f(x)=x^2+36 x+12$
B. $f(x)=x^2-36 x-12$
C. $f(x)=-x^2+12 x+36$
D. $f(x)=-x^2-12 x-36$
Answer (1)
The parabola touches the x-axis at x = − 6 , implying the quadratic function has the form f ( x ) = a ( x + 6 ) 2 .
Expanding the general form gives f ( x ) = a x 2 + 12 a x + 36 a .
Check each option to see if it matches the general form.
The correct function is f ( x ) = − x 2 − 12 x − 36 , which can be written as − ( x + 6 ) 2 , so the final answer is f ( x ) = − x 2 − 12 x − 36 .
Explanation
Understanding the Problem We are looking for a quadratic function that represents a parabola that touches the x -axis at x = − 6 . This means the parabola has a vertex on the x -axis at x = − 6 . Therefore, the quadratic function must have the form f ( x ) = a ( x + 6 ) 2 for some non-zero constant a . Expanding this, we get f ( x ) = a ( x 2 + 12 x + 36 ) = a x 2 + 12 a x + 36 a . We need to find which of the given options matches this form.
Checking Each Option Let's examine each option:
f ( x ) = x 2 + 36 x + 12 : For this to be of the form a ( x + 6 ) 2 , we would need a = 1 , so f ( x ) = x 2 + 12 x + 36 . But the given function is x 2 + 36 x + 12 , so this is not the correct function.
f ( x ) = x 2 − 36 x − 12 : This also cannot be of the form a ( x + 6 ) 2 because the coefficient of x is negative.
f ( x ) = − x 2 + 12 x + 36 : For this to be of the form a ( x + 6 ) 2 , we would need a = − 1 , so f ( x ) = − x 2 − 12 x − 36 . But the given function is − x 2 + 12 x + 36 , so this is not the correct function.
f ( x ) = − x 2 − 12 x − 36 : This can be written as f ( x ) = − ( x 2 + 12 x + 36 ) = − ( x + 6 ) 2 . So, a = − 1 . This is the correct function.
Final Answer Therefore, the quadratic function that represents a parabola that touches the x -axis at x = − 6 is f ( x ) = − x 2 − 12 x − 36 .
Examples
Understanding quadratic functions and their properties, such as the vertex form, is crucial in various real-world applications. For instance, designing a parabolic reflector for a flashlight or a satellite dish requires knowing how the coefficients of the quadratic function affect the shape and position of the parabola. Similarly, in physics, projectile motion can be modeled using quadratic functions, where the vertex represents the maximum height reached by the projectile. By mastering these concepts, one can optimize designs and predict outcomes in diverse fields.
