Question
Which function describes this graph?
26
10
lath
-一一一
-101
10
-10|
0y=x*+8x+12
0y=x'-2x+6
0 y=(x-4)(x-4)
O y =(x-2)(x-6)
26
10
lath
-一一一
-101
10
-10|
0y=x*+8x+12
0y=x'-2x+6
0 y=(x-4)(x-4)
O y =(x-2)(x-6)
Answers
Answered by
GPT-4o mini
To determine which function describes the graph, we need to analyze the given function options:
1. **\(y = x^2 + 8x + 12\)**: This is a quadratic function in the standard form. We can factor it to find its roots if needed.
2. **\(y = x^2 - 2x + 6\)**: Another quadratic function, we can analyze it similarly.
3. **\(y = (x-4)(x-4)\)**: This is a perfect square trinomial, which expands to \(y = x^2 - 8x + 16\).
4. **\(y = (x-2)(x-6)\)**: This is also a quadratic expression that expands to \(y = x^2 - 8x + 12\).
### Let's explore the factorizations:
1. **\(y = x^2 + 8x + 12\)**
- Roots: To find the roots, we can use the factoring or quadratic formula; however, for graphing, we may just look for vertex and intercepts.
2. **\(y = x^2 - 2x + 6\)**
- Roots: This equation has no real roots since the discriminant \((-2)^2 - 4(1)(6) < 0\). This parabola opens upwards and does not cross the x-axis.
3. **\(y = (x-4)(x-4)\)**
- Roots: This has a double root at \(x = 4\).
- Vertex: The vertex is at \(x = 4\) and opens upwards.
4. **\(y = (x-2)(x-6)\)**
- Roots: It crosses the x-axis at \(x = 2\) and \(x = 6\), finding the intercepts and shapes a parabola.
### Graphical behavior:
- If the graph you are describing crosses the x-axis or has clear intercepts, options 3 and 4 (both quadratic forms) would be potential candidates, especially option 4 if it crosses x-axis.
If you provide more information regarding the shape of the graph (does it open upwards or downwards, where it intersects the axes), we can confirm which equation represents the correct graph. However, based on the options, the one that crosses the axes is likely:
**\(y = (x-2)(x-6)\)**, since this one gives us real roots and a graph that intersects the x-axis at points x=2 and x=6.
1. **\(y = x^2 + 8x + 12\)**: This is a quadratic function in the standard form. We can factor it to find its roots if needed.
2. **\(y = x^2 - 2x + 6\)**: Another quadratic function, we can analyze it similarly.
3. **\(y = (x-4)(x-4)\)**: This is a perfect square trinomial, which expands to \(y = x^2 - 8x + 16\).
4. **\(y = (x-2)(x-6)\)**: This is also a quadratic expression that expands to \(y = x^2 - 8x + 12\).
### Let's explore the factorizations:
1. **\(y = x^2 + 8x + 12\)**
- Roots: To find the roots, we can use the factoring or quadratic formula; however, for graphing, we may just look for vertex and intercepts.
2. **\(y = x^2 - 2x + 6\)**
- Roots: This equation has no real roots since the discriminant \((-2)^2 - 4(1)(6) < 0\). This parabola opens upwards and does not cross the x-axis.
3. **\(y = (x-4)(x-4)\)**
- Roots: This has a double root at \(x = 4\).
- Vertex: The vertex is at \(x = 4\) and opens upwards.
4. **\(y = (x-2)(x-6)\)**
- Roots: It crosses the x-axis at \(x = 2\) and \(x = 6\), finding the intercepts and shapes a parabola.
### Graphical behavior:
- If the graph you are describing crosses the x-axis or has clear intercepts, options 3 and 4 (both quadratic forms) would be potential candidates, especially option 4 if it crosses x-axis.
If you provide more information regarding the shape of the graph (does it open upwards or downwards, where it intersects the axes), we can confirm which equation represents the correct graph. However, based on the options, the one that crosses the axes is likely:
**\(y = (x-2)(x-6)\)**, since this one gives us real roots and a graph that intersects the x-axis at points x=2 and x=6.