Answer :
To determine which graph represents the equation [tex]\( y = -(x - 1)^2 + 1 \)[/tex], let's go through a step-by-step analysis.
### Step 1: Understand the Form of the Equation
The given equation is a quadratic equation in the form:
[tex]\[ y = a(x - h)^2 + k \][/tex]
where [tex]\( a = -1 \)[/tex], [tex]\( h = 1 \)[/tex], and [tex]\( k = 1 \)[/tex].
### Step 2: Identify the Vertex
The vertex of a parabola given by the equation [tex]\( y = a(x - h)^2 + k \)[/tex] is located at the point [tex]\( (h, k) \)[/tex].
Therefore, the vertex of the given equation [tex]\( y = -(x - 1)^2 + 1 \)[/tex] is at:
[tex]\[ (h, k) = (1, 1) \][/tex]
### Step 3: Determine the Direction of the Parabola
The coefficient [tex]\( a \)[/tex] in front of the squared term determines the direction in which the parabola opens:
- If [tex]\( a > 0 \)[/tex], the parabola opens upwards.
- If [tex]\( a < 0 \)[/tex], the parabola opens downwards.
Since [tex]\( a = -1 \)[/tex], which is less than 0, this parabola opens downwards.
### Step 4: Sketch and Analyze the Parabola
Based on the vertex and the direction, let's sketch the parabola:
- The vertex is at [tex]\( (1, 1) \)[/tex].
- The parabola opens downwards.
The general shape of this parabola would look like an inverted "U" shape centered at the vertex.
### Step 5: Confirm the Shape by Plugging in Values (Optional Check)
For additional confirmation, you can plug in a few values of [tex]\( x \)[/tex] to find corresponding values of [tex]\( y \)[/tex] and plot those points:
1. When [tex]\( x = 1 \)[/tex]:
[tex]\[ y = -(1 - 1)^2 + 1 = 1 \][/tex]
Point: [tex]\( (1, 1) \)[/tex]
2. When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -(0 - 1)^2 + 1 = -(1) + 1 = 0 \][/tex]
Point: [tex]\( (0, 0) \)[/tex]
3. When [tex]\( x = 2 \)[/tex]:
[tex]\[ y = -(2 - 1)^2 + 1 = -(1) + 1 = 0 \][/tex]
Point: [tex]\( (2, 0) \)[/tex]
4. When [tex]\( x = -1 \)[/tex]:
[tex]\[ y = -(-1 - 1)^2 + 1 = -4 + 1 = -3 \][/tex]
Point: [tex]\( (-1, -3) \)[/tex]
These points should lie on the parabola, and you can connect them to visualize its shape.
### Conclusion
The graph representing the equation [tex]\( y = -(x - 1)^2 + 1 \)[/tex] is a downward-opening parabola with its vertex located at [tex]\( (1, 1) \)[/tex]. It is symmetric around the vertical line [tex]\( x = 1 \)[/tex] and will intersect the [tex]\( y \)[/tex]-axis somewhere below the vertex.
### Step 1: Understand the Form of the Equation
The given equation is a quadratic equation in the form:
[tex]\[ y = a(x - h)^2 + k \][/tex]
where [tex]\( a = -1 \)[/tex], [tex]\( h = 1 \)[/tex], and [tex]\( k = 1 \)[/tex].
### Step 2: Identify the Vertex
The vertex of a parabola given by the equation [tex]\( y = a(x - h)^2 + k \)[/tex] is located at the point [tex]\( (h, k) \)[/tex].
Therefore, the vertex of the given equation [tex]\( y = -(x - 1)^2 + 1 \)[/tex] is at:
[tex]\[ (h, k) = (1, 1) \][/tex]
### Step 3: Determine the Direction of the Parabola
The coefficient [tex]\( a \)[/tex] in front of the squared term determines the direction in which the parabola opens:
- If [tex]\( a > 0 \)[/tex], the parabola opens upwards.
- If [tex]\( a < 0 \)[/tex], the parabola opens downwards.
Since [tex]\( a = -1 \)[/tex], which is less than 0, this parabola opens downwards.
### Step 4: Sketch and Analyze the Parabola
Based on the vertex and the direction, let's sketch the parabola:
- The vertex is at [tex]\( (1, 1) \)[/tex].
- The parabola opens downwards.
The general shape of this parabola would look like an inverted "U" shape centered at the vertex.
### Step 5: Confirm the Shape by Plugging in Values (Optional Check)
For additional confirmation, you can plug in a few values of [tex]\( x \)[/tex] to find corresponding values of [tex]\( y \)[/tex] and plot those points:
1. When [tex]\( x = 1 \)[/tex]:
[tex]\[ y = -(1 - 1)^2 + 1 = 1 \][/tex]
Point: [tex]\( (1, 1) \)[/tex]
2. When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -(0 - 1)^2 + 1 = -(1) + 1 = 0 \][/tex]
Point: [tex]\( (0, 0) \)[/tex]
3. When [tex]\( x = 2 \)[/tex]:
[tex]\[ y = -(2 - 1)^2 + 1 = -(1) + 1 = 0 \][/tex]
Point: [tex]\( (2, 0) \)[/tex]
4. When [tex]\( x = -1 \)[/tex]:
[tex]\[ y = -(-1 - 1)^2 + 1 = -4 + 1 = -3 \][/tex]
Point: [tex]\( (-1, -3) \)[/tex]
These points should lie on the parabola, and you can connect them to visualize its shape.
### Conclusion
The graph representing the equation [tex]\( y = -(x - 1)^2 + 1 \)[/tex] is a downward-opening parabola with its vertex located at [tex]\( (1, 1) \)[/tex]. It is symmetric around the vertical line [tex]\( x = 1 \)[/tex] and will intersect the [tex]\( y \)[/tex]-axis somewhere below the vertex.
