Determining a Pemester 1\( ) \) ic The graph of \( f(x)=|x| \) has been stretched by a factor of 2.5 . If no other transformations of the function have occurred, which point lies on the new graph? \( (-4,-4) \) \( (-3,7.5) \) \( (-2,5.5) \) \( (-1,-2.5) \)

To determine which point lies on the new graph after stretching \( f(x) = |x| \) by a factor of 2.5, we need to understand how the transformation affects the function. ### Understanding the Transformation - **Vertical Stretch**: When a function is stretched vertically by a factor of \( a \), the new function becomes \( g(x) = a \cdot f(x) \). This means each \( y \)-coordinate of the original function is multiplied by \( a \). Given that \( f(x) = |x| \) is stretched vertically by a factor of 2.5, the new function will be: \[ g(x) = 2.5 \cdot |x| \] ### Checking the Points Let's evaluate \( g(x) \) for each \( x \)-value in the given points: 1. **Point \((-4, -4)\)** \[ g(-4) = 2.5 \cdot |-4| = 2.5 \cdot 4 = 10 \neq -4 \] This point **does not** lie on the new graph. 2. **Point \((-3, 7.5)\)** \[ g(-3) = 2.5 \cdot |-3| = 2.5 \cdot 3 = 7.5 \] This point **does** lie on the new graph. 3. **Point \((-2, 5.5)\)** \[ g(-2) = 2.5 \cdot |-2| = 2.5 \cdot 2 = 5 \neq 5.5 \] This point **does not** lie on the new graph. 4. **Point \((-1, -2.5)\)** \[ g(-1) = 2.5 \cdot |-1| = 2.5 \cdot 1 = 2.5 \neq -2.5 \] This point **does not** lie on the new graph. ### Conclusion After evaluating all the points, only **\((-3, 7.5)\)** satisfies the condition of the transformed function. **Answer:** \( (-3,\ 7.5) \)