Question
Sketch a graph with a line of symmetry at x = -3, a range of [1, infinity) and a y-intercept of 10 and find the equation
Answers
Answered by
GPT-4o mini
To create a graph with the specified properties, we need to define a function that meets the following criteria:
1. **Line of symmetry at x = -3**: This means if you reflect the graph across the line \( x = -3 \), it should look the same.
2. **Range of \([1, \infty)\)**: The output (y-values) of the function should be at least 1 or greater.
3. **Y-intercept of 10**: The value of the function when \( x = 0 \) should be 10.
### Step 1: Symmetry
To have a line of symmetry at \( x = -3 \), we can express the function in terms of a variable shifted by -3. Let's use \( h(x) = f(x + 3) \).
### Step 2: Range
For the range to be \([1, \infty)\), we need to ensure our function has a minimum value of 1. A simple choice would be a parabola that opens upwards.
### Step 3: Y-intercept
We need the y-intercept to be 10. Since we want a function of the form:
\[ y = a(x + 3)^2 + k \]
where \( k \) will determine the minimum (or vertex) value. If this function takes a minimum of \( k = 1 \), we can get:
\[ y = a(x + 3)^2 + 1 \]
To find \( a \), we can set \( x = 0 \) to ensure the function has a y-intercept of 10:
\[ 10 = a(0 + 3)^2 + 1 \]
\[ 10 = 9a + 1 \]
\[ 9 = 9a \]
\[ a = 1 \]
### Final Equation
Thus, our function becomes:
\[ y = (x + 3)^2 + 1 \]
### Validation
1. **Y-intercept**:
\[ y(0) = (0 + 3)^2 + 1 = 9 + 1 = 10 \]
Correct y-intercept.
2. **Line of symmetry**:
The vertex is at \( x = -3 \), so the line of symmetry is at \( x = -3 \).
3. **Range**:
As the minimum value is 1 (when \( x = -3 \)) and the parabola opens upwards, the range is \([1, \infty)\).
### Graph sketch
To sketch the graph, plot the vertex at \((-3, 1)\) and the y-intercept at \((0, 10)\), drawing a symmetrical parabola opening upwards through these points.
### Conclusion
The equation of the function that meets all the specified criteria is:
\[
y = (x + 3)^2 + 1
\]
1. **Line of symmetry at x = -3**: This means if you reflect the graph across the line \( x = -3 \), it should look the same.
2. **Range of \([1, \infty)\)**: The output (y-values) of the function should be at least 1 or greater.
3. **Y-intercept of 10**: The value of the function when \( x = 0 \) should be 10.
### Step 1: Symmetry
To have a line of symmetry at \( x = -3 \), we can express the function in terms of a variable shifted by -3. Let's use \( h(x) = f(x + 3) \).
### Step 2: Range
For the range to be \([1, \infty)\), we need to ensure our function has a minimum value of 1. A simple choice would be a parabola that opens upwards.
### Step 3: Y-intercept
We need the y-intercept to be 10. Since we want a function of the form:
\[ y = a(x + 3)^2 + k \]
where \( k \) will determine the minimum (or vertex) value. If this function takes a minimum of \( k = 1 \), we can get:
\[ y = a(x + 3)^2 + 1 \]
To find \( a \), we can set \( x = 0 \) to ensure the function has a y-intercept of 10:
\[ 10 = a(0 + 3)^2 + 1 \]
\[ 10 = 9a + 1 \]
\[ 9 = 9a \]
\[ a = 1 \]
### Final Equation
Thus, our function becomes:
\[ y = (x + 3)^2 + 1 \]
### Validation
1. **Y-intercept**:
\[ y(0) = (0 + 3)^2 + 1 = 9 + 1 = 10 \]
Correct y-intercept.
2. **Line of symmetry**:
The vertex is at \( x = -3 \), so the line of symmetry is at \( x = -3 \).
3. **Range**:
As the minimum value is 1 (when \( x = -3 \)) and the parabola opens upwards, the range is \([1, \infty)\).
### Graph sketch
To sketch the graph, plot the vertex at \((-3, 1)\) and the y-intercept at \((0, 10)\), drawing a symmetrical parabola opening upwards through these points.
### Conclusion
The equation of the function that meets all the specified criteria is:
\[
y = (x + 3)^2 + 1
\]