Answer :
To analyze the function [tex]\( f(x) = (x+4)(x-6) \)[/tex], let's go through the key points that will allow us to determine the correct statements.
### Step-by-Step Analysis:
1. Form of the Function:
The function is given as [tex]\( f(x) = (x+4)(x-6) \)[/tex].
2. Expansion of the Function:
Expanding [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = (x+4)(x-6) = x^2 - 2x - 24 \][/tex]
3. Vertex of the Function:
The vertex form of a quadratic function [tex]\( f(x) = ax^2 + bx + c \)[/tex] gives us the x-coordinate of the vertex as [tex]\( x = -\frac{b}{2a} \)[/tex].
Given [tex]\( a = 1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = -24 \)[/tex]:
[tex]\[ x_{\text{vertex}} = -\frac{-2}{2 \cdot 1} = 1 \][/tex]
To find the y-coordinate of the vertex, substitute [tex]\( x = 1 \)[/tex] back into the function:
[tex]\[ y_{\text{vertex}} = (1^2) - 2(1) - 24 = 1 - 2 - 24 = -25 \][/tex]
Therefore, the vertex of the function is at [tex]\( (1, -25) \)[/tex].
4. Intervals Where the Function is Increasing/Decreasing:
Since the coefficient of [tex]\( x^2 \)[/tex] (which is [tex]\( a = 1 \)[/tex]) is positive, the parabola opens upwards. This means:
- The function is decreasing to the left of the vertex (i.e., for [tex]\( x < 1 \)[/tex])
- The function is increasing to the right of the vertex (i.e., for [tex]\( x > 1 \)[/tex])
5. Intervals Where the Function Is Positive/Negative:
Finding the roots of the quadratic equation [tex]\( x^2 - 2x - 24 = 0 \)[/tex]:
[tex]\[ (x+4)(x-6) = 0 \][/tex]
Thus, the solutions are [tex]\( x = -4 \)[/tex] and [tex]\( x = 6 \)[/tex].
- The function is positive on the intervals [tex]\( (-\infty, -4) \)[/tex] and [tex]\( (6, \infty) \)[/tex]
- The function is negative on the interval [tex]\( (-4, 6) \)[/tex]
### Conclusion:
Given the above analysis, the correct statements are:
1. The vertex of the function is at [tex]\( (1, -25) \)[/tex]. This is a true statement, as determined from the vertex calculation.
2. The graph is negative on the entire interval [tex]\( -4 < x < 6 \)[/tex]. This is true because the function is below the x-axis (negative) between its roots [tex]\( x = -4 \)[/tex] and [tex]\( x = 6 \)[/tex].
The other statements do not correctly describe the properties of the function.
### Answer:
- The vertex of the function is at [tex]\( (1, -25) \)[/tex].
- The graph is negative on the entire interval [tex]\( -4 < x < 6 \)[/tex].
### Step-by-Step Analysis:
1. Form of the Function:
The function is given as [tex]\( f(x) = (x+4)(x-6) \)[/tex].
2. Expansion of the Function:
Expanding [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = (x+4)(x-6) = x^2 - 2x - 24 \][/tex]
3. Vertex of the Function:
The vertex form of a quadratic function [tex]\( f(x) = ax^2 + bx + c \)[/tex] gives us the x-coordinate of the vertex as [tex]\( x = -\frac{b}{2a} \)[/tex].
Given [tex]\( a = 1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = -24 \)[/tex]:
[tex]\[ x_{\text{vertex}} = -\frac{-2}{2 \cdot 1} = 1 \][/tex]
To find the y-coordinate of the vertex, substitute [tex]\( x = 1 \)[/tex] back into the function:
[tex]\[ y_{\text{vertex}} = (1^2) - 2(1) - 24 = 1 - 2 - 24 = -25 \][/tex]
Therefore, the vertex of the function is at [tex]\( (1, -25) \)[/tex].
4. Intervals Where the Function is Increasing/Decreasing:
Since the coefficient of [tex]\( x^2 \)[/tex] (which is [tex]\( a = 1 \)[/tex]) is positive, the parabola opens upwards. This means:
- The function is decreasing to the left of the vertex (i.e., for [tex]\( x < 1 \)[/tex])
- The function is increasing to the right of the vertex (i.e., for [tex]\( x > 1 \)[/tex])
5. Intervals Where the Function Is Positive/Negative:
Finding the roots of the quadratic equation [tex]\( x^2 - 2x - 24 = 0 \)[/tex]:
[tex]\[ (x+4)(x-6) = 0 \][/tex]
Thus, the solutions are [tex]\( x = -4 \)[/tex] and [tex]\( x = 6 \)[/tex].
- The function is positive on the intervals [tex]\( (-\infty, -4) \)[/tex] and [tex]\( (6, \infty) \)[/tex]
- The function is negative on the interval [tex]\( (-4, 6) \)[/tex]
### Conclusion:
Given the above analysis, the correct statements are:
1. The vertex of the function is at [tex]\( (1, -25) \)[/tex]. This is a true statement, as determined from the vertex calculation.
2. The graph is negative on the entire interval [tex]\( -4 < x < 6 \)[/tex]. This is true because the function is below the x-axis (negative) between its roots [tex]\( x = -4 \)[/tex] and [tex]\( x = 6 \)[/tex].
The other statements do not correctly describe the properties of the function.
### Answer:
- The vertex of the function is at [tex]\( (1, -25) \)[/tex].
- The graph is negative on the entire interval [tex]\( -4 < x < 6 \)[/tex].
