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We are given the function [tex]\( f(x) = (x+4)(x-6) \)[/tex] and some statements. Let's discuss each of the statements step by step to determine which ones are true.
1. The vertex of the function is at [tex]\((1, -25)\)[/tex].
To find the vertex of a quadratic function [tex]\( f(x) = ax^2 + bx + c \)[/tex], we use the formula for the x-coordinate of the vertex, [tex]\( x = -\frac{b}{2a} \)[/tex]. For our function [tex]\( f(x) \)[/tex], the function can be rewritten as:
[tex]\[ f(x) = x^2 - 2x - 24 \][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = -24 \)[/tex]. The x-coordinate of the vertex is:
[tex]\[ x = -\frac{-2}{2 \cdot 1} = 1 \][/tex]
Plugging [tex]\( x = 1 \)[/tex] back into the function to find the y-coordinate:
[tex]\[ f(1) = (1 + 4)(1 - 6) = 5 \cdot (-5) = -25 \][/tex]
Hence, the vertex is indeed at [tex]\( (1, -25) \)[/tex]. This statement is true.
2. The vertex of the function is at [tex]\((1, -24)\)[/tex].
As found above, the vertex is at [tex]\( (1, -25) \)[/tex], not [tex]\( (1, -24) \)[/tex]. Therefore, this statement is false.
3. The graph is increasing only on the interval [tex]\(-4 < x < 6\)[/tex].
For a quadratic function that opens upwards (as seen by the positive [tex]\( a \)[/tex] value), the function is decreasing before the vertex and increasing after the vertex. Since our vertex is at [tex]\( x = 1 \)[/tex]:
[tex]\[ \text{The graph is increasing for } x > 1 \][/tex]
Therefore, this statement is false.
4. The graph is positive only on one interval, where [tex]\( x < -4 \)[/tex].
To find where the graph is positive or negative, we need to identify the roots of the function [tex]\( f(x) = (x + 4)(x - 6) \)[/tex]. The roots are:
[tex]\[ x = -4 \quad \text{and} \quad x = 6 \][/tex]
The function is positive outside the interval between the roots, so it is positive for [tex]\( x < -4 \)[/tex] and [tex]\( x > 6 \)[/tex]. Therefore, the statement that the graph is positive only on one interval is false because there are two intervals [tex]\( (-\infty, -4) \)[/tex] and [tex]\( (6, \infty) \)[/tex].
5. The graph is negative on the entire interval [tex]\(-4 < x < 6\)[/tex].
As identified, the function [tex]\( f(x) = (x + 4)(x - 6) \)[/tex] is negative between its roots [tex]\( -4 \)[/tex] and [tex]\( 6 \)[/tex]. Thus, this statement is true.
Given our analysis, the two correct statements are:
- The vertex of the function is at [tex]\((1, -25)\)[/tex].
- The graph is negative on the entire interval [tex]\(-4 < x < 6\)[/tex].
1. The vertex of the function is at [tex]\((1, -25)\)[/tex].
To find the vertex of a quadratic function [tex]\( f(x) = ax^2 + bx + c \)[/tex], we use the formula for the x-coordinate of the vertex, [tex]\( x = -\frac{b}{2a} \)[/tex]. For our function [tex]\( f(x) \)[/tex], the function can be rewritten as:
[tex]\[ f(x) = x^2 - 2x - 24 \][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = -24 \)[/tex]. The x-coordinate of the vertex is:
[tex]\[ x = -\frac{-2}{2 \cdot 1} = 1 \][/tex]
Plugging [tex]\( x = 1 \)[/tex] back into the function to find the y-coordinate:
[tex]\[ f(1) = (1 + 4)(1 - 6) = 5 \cdot (-5) = -25 \][/tex]
Hence, the vertex is indeed at [tex]\( (1, -25) \)[/tex]. This statement is true.
2. The vertex of the function is at [tex]\((1, -24)\)[/tex].
As found above, the vertex is at [tex]\( (1, -25) \)[/tex], not [tex]\( (1, -24) \)[/tex]. Therefore, this statement is false.
3. The graph is increasing only on the interval [tex]\(-4 < x < 6\)[/tex].
For a quadratic function that opens upwards (as seen by the positive [tex]\( a \)[/tex] value), the function is decreasing before the vertex and increasing after the vertex. Since our vertex is at [tex]\( x = 1 \)[/tex]:
[tex]\[ \text{The graph is increasing for } x > 1 \][/tex]
Therefore, this statement is false.
4. The graph is positive only on one interval, where [tex]\( x < -4 \)[/tex].
To find where the graph is positive or negative, we need to identify the roots of the function [tex]\( f(x) = (x + 4)(x - 6) \)[/tex]. The roots are:
[tex]\[ x = -4 \quad \text{and} \quad x = 6 \][/tex]
The function is positive outside the interval between the roots, so it is positive for [tex]\( x < -4 \)[/tex] and [tex]\( x > 6 \)[/tex]. Therefore, the statement that the graph is positive only on one interval is false because there are two intervals [tex]\( (-\infty, -4) \)[/tex] and [tex]\( (6, \infty) \)[/tex].
5. The graph is negative on the entire interval [tex]\(-4 < x < 6\)[/tex].
As identified, the function [tex]\( f(x) = (x + 4)(x - 6) \)[/tex] is negative between its roots [tex]\( -4 \)[/tex] and [tex]\( 6 \)[/tex]. Thus, this statement is true.
Given our analysis, the two correct statements are:
- The vertex of the function is at [tex]\((1, -25)\)[/tex].
- The graph is negative on the entire interval [tex]\(-4 < x < 6\)[/tex].
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