To identify the correct graph representing the function [tex]\( y = x^2 - 6x - 16 \)[/tex], follow these detailed steps:
### Step 1: Identify Characteristics of the Quadratic Function
The given function is a quadratic function of the form [tex]\( y = ax^2 + bx + c \)[/tex], where:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -6 \)[/tex]
- [tex]\( c = -16 \)[/tex]
Quadratic functions are parabolas. Since [tex]\( a > 0 \)[/tex], the parabola opens upwards.
### Step 2: Find the Vertex of the Parabola
The vertex of a parabola [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the formula for the x-coordinate of the vertex:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Substitute [tex]\( a = 1 \)[/tex] and [tex]\( b = -6 \)[/tex] into the formula:
[tex]\[ x = -\frac{-6}{2 \times 1} = \frac{6}{2} = 3 \][/tex]
To find the y-coordinate of the vertex, substitute [tex]\( x = 3 \)[/tex] back into the function:
[tex]\[ y = (3)^2 - 6(3) - 16 \][/tex]
[tex]\[ y = 9 - 18 - 16 \][/tex]
[tex]\[ y = -25 \][/tex]
So, the vertex of the parabola is at [tex]\( (3, -25) \)[/tex].
### Step 3: Confirm Additional Key Points
For additional assurance, let's determine the y-intercept of the function by evaluating it at [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 0^2 - 6(0) - 16 \][/tex]
[tex]\[ y = -16 \][/tex]
The y-intercept is [tex]\( (0, -16) \)[/tex].
### Step 4: Analyze the Graphs
- Look for a graph that matches the determined vertex at [tex]\( (3, -25) \)[/tex].
- The graph should open upwards since [tex]\( a = 1 \)[/tex].
- The y-intercept should be at [tex]\( (0, -16) \)[/tex].
### Conclusion
The graph that meets the above conditions (vertex at [tex]\( (3, -25) \)[/tex], opens upwards, y-intercept at [tex]\( (0, -16) \)[/tex]) is the correct graph representing the function [tex]\( y = x^2 - 6x - 16 \)[/tex].
Without the actual graphs, based on the detailed analysis, identify the graph that corresponds correctly to the described characteristics to choose the correct option.
