Question
Which graph represents the function f(x) = (x – 3)2?
On a coordinate plane, a parabola opens up. It goes through (negative 6, 8), has a vertex at (negative 3, 0), and goes through (negative 0, 8).
On a coordinate plane, a parabola opens up. It goes through (negative 4, 7), has a vertex at (0, negative 9), and goes through (4, 7).
On a coordinate plane, a parabola opens up. It goes through (1, 4), has a vertex at (3, 0), and goes through (5, 4).
On a coordinate plane, a parabola opens up. It goes through (negative 2, 7), has a vertex at (0, 3), and goes through (2, 7).
On a coordinate plane, a parabola opens up. It goes through (negative 6, 8), has a vertex at (negative 3, 0), and goes through (negative 0, 8).
On a coordinate plane, a parabola opens up. It goes through (negative 4, 7), has a vertex at (0, negative 9), and goes through (4, 7).
On a coordinate plane, a parabola opens up. It goes through (1, 4), has a vertex at (3, 0), and goes through (5, 4).
On a coordinate plane, a parabola opens up. It goes through (negative 2, 7), has a vertex at (0, 3), and goes through (2, 7).
Answers
GPT-4o mini
To determine which graph represents the function \( f(x) = (x - 3)^2 \), we start by analyzing the characteristics of this quadratic function.
1. **Vertex**: The vertex of the function \( f(x) = (x - 3)^2 \) is at the point \( (3, 0) \).
2. **Direction**: The parabola opens upwards since the coefficient of the squared term is positive.
3. **Intercepts**: The y-intercept can be found by substituting \( x = 0 \):
\[
f(0) = (0 - 3)^2 = 9 \text{.}
\]
So the function passes through the point \( (0, 9) \).
4. **Symmetry**: The parabola is symmetric around the vertical line \( x = 3 \).
Now, we can evaluate the given options based on these characteristics:
- **First graph**: Vertex at \((-3, 0)\), doesn’t match.
- **Second graph**: Vertex at \((0, -9)\), doesn’t match.
- **Third graph**: Vertex at \((3, 0)\) and passes through points \( (1, 4) \) and \( (5, 4) \), seems like it could match.
- **Fourth graph**: Vertex at \((0, 3)\), doesn’t match.
Now, let's verify the points for the third graph:
- The vertex is \((3, 0)\), which is correct.
- To see if it goes through \((1, 4)\):
\[
f(1) = (1 - 3)^2 = (-2)^2 = 4 \text{, which is correct.}
\]
- The point \((5, 4)\):
\[
f(5) = (5 - 3)^2 = (2)^2 = 4 \text{, which is also correct.}
\]
Since the third graph correctly fits all criteria:
The correct graph representing the function \( f(x) = (x - 3)^2 \) is the **third graph**, which has the vertex at \( (3, 0) \) and passes through \( (1, 4) \) and \( (5, 4) \).
1. **Vertex**: The vertex of the function \( f(x) = (x - 3)^2 \) is at the point \( (3, 0) \).
2. **Direction**: The parabola opens upwards since the coefficient of the squared term is positive.
3. **Intercepts**: The y-intercept can be found by substituting \( x = 0 \):
\[
f(0) = (0 - 3)^2 = 9 \text{.}
\]
So the function passes through the point \( (0, 9) \).
4. **Symmetry**: The parabola is symmetric around the vertical line \( x = 3 \).
Now, we can evaluate the given options based on these characteristics:
- **First graph**: Vertex at \((-3, 0)\), doesn’t match.
- **Second graph**: Vertex at \((0, -9)\), doesn’t match.
- **Third graph**: Vertex at \((3, 0)\) and passes through points \( (1, 4) \) and \( (5, 4) \), seems like it could match.
- **Fourth graph**: Vertex at \((0, 3)\), doesn’t match.
Now, let's verify the points for the third graph:
- The vertex is \((3, 0)\), which is correct.
- To see if it goes through \((1, 4)\):
\[
f(1) = (1 - 3)^2 = (-2)^2 = 4 \text{, which is correct.}
\]
- The point \((5, 4)\):
\[
f(5) = (5 - 3)^2 = (2)^2 = 4 \text{, which is also correct.}
\]
Since the third graph correctly fits all criteria:
The correct graph representing the function \( f(x) = (x - 3)^2 \) is the **third graph**, which has the vertex at \( (3, 0) \) and passes through \( (1, 4) \) and \( (5, 4) \).