Which function does the graph represent?
A. f(x) = (x + 3)(x + 7)
B. f(x) = (x – 3)(x – 7)
C. f(x) = 3(x – 3)(x – 7)
D. f(x) = 11(x + 3)(x + 7)
Answer (2)
To determine which function the graph represents, we need to understand how the functions provided are structured and what their implications are. The functions given are all quadratic equations factored into binomials.
Understanding the options:
Option 1: f ( x ) = ( x + 3 ) ( x + 7 )
Option 2: f ( x ) = ( x − 3 ) ( x − 7 )
Option 3: f ( x ) = 3 ( x − 3 ) ( x − 7 )
Option 4: f ( x ) = 11 ( x + 3 ) ( x + 7 )
Identify the key features: Each function can be expanded to a standard quadratic form a x 2 + b x + c , where the sign and value of a as well as the roots (where f ( x ) = 0 ) are important features.
Roots of the functions:
Option 1 has roots at x = − 3 and x = − 7 .
Option 2 has roots at x = 3 and x = 7 .
Option 3 has roots at x = 3 and x = 7 .
Option 4 has roots at x = − 3 and x = − 7 .
Finding the correct function:
If the graph has roots (x-intercepts) at, for example, x = 3 and x = 7 , then Options 2 and 3 match this description regarding roots.
Option 3 is a scaled version of Option 2 , having a leading coefficient of 3. This means the graph would open in the same direction but be stretched vertically due to the factor of 3.
Conclusion: To correctly match a graph to one of these equations, we must verify if the graph's intercepts, direction of opening, and stretching match one of our function options. If, let's say, the parabola opens upwards (which they all do here) and has intercepts at x = 3 and x = 7 but has an additional vertical stretch, Option 3 f ( x ) = 3 ( x − 3 ) ( x − 7 ) would be the correct choice.
To identify the function represented by the graph, we examine the roots from the given equations. If the graph has x-intercepts at x = 3 and x = 7 and is vertically stretched, then the correct choice is Option C: f(x) = 3(x – 3)(x – 7).
;
