Chapter9: Quadratic Equations And Functions
Section9.7: Graph Quadratic Functions Using Transformations
Problem 362E: Graph the quadratic function f(x)=2x24x3 first using the properties as we did in the last section...
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Use transformations- problem attached. Show all work
![**Graphing Quadratic Functions and Determining Domain and Range**
In this lesson, we will graph the function \( y = (x - 3)^2 - 4 \) using transformations, and we will state the domain and range using interval notation.
### Step-by-Step Instructions:
1. **Identify the Parent Function**:
The parent function of \( y = (x - 3)^2 - 4 \) is \( y = x^2 \).
2. **Apply Horizontal Shift**:
The function \( y = (x - 3)^2 \) represents a horizontal shift of the parent function \( y = x^2 \) to the right by 3 units.
3. **Apply Vertical Shift**:
The function \( y = (x - 3)^2 - 4 \) represents a vertical shift of the function \( y = (x - 3)^2 \) downward by 4 units.
### Graphing the Function:
1. **Draw the Coordinate Axes**:
Start by drawing the x-axis and y-axis with suitable scales. In the given image, the axes range from -5 to 5 on both scales.
2. **Plot the Vertex**:
The vertex of the function \( y = (x - 3)^2 - 4 \) is at the point (3, -4).
3. **Sketch the Parabola**:
From the vertex (3, -4), sketch the parabola opening upwards. Use symmetry about the vertex and plot additional points symmetrically on either side of the vertex for more accuracy.
### Explanation of the Given Graph:
The provided graph is a coordinate plane with the x-axis ranging from -5 to 5 and the y-axis ranging from -5 to 5. The graph is currently empty, ready for you to plot the given quadratic function using the transformations described above.
### Domain and Range:
**Domain**: The domain of \( y = (x - 3)^2 - 4 \) includes all real numbers, as the function is defined for all x-values. Thus, the domain is:
\[ \text{Domain: } (-\infty, \infty) \]
**Range**: The range of the function is determined by the vertex, which is the lowest point on the graph. Since the vertex is at (3, -4) and the parabola opens upwards, the y-values](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F29666072-4841-4557-93a0-541aeee2aafd%2Fcff62a91-7a87-4e40-8e0b-6d364dafbc3d%2F85tuix_processed.png&w=3840&q=75)
Transcribed Image Text:**Graphing Quadratic Functions and Determining Domain and Range**
In this lesson, we will graph the function \( y = (x - 3)^2 - 4 \) using transformations, and we will state the domain and range using interval notation.
### Step-by-Step Instructions:
1. **Identify the Parent Function**:
The parent function of \( y = (x - 3)^2 - 4 \) is \( y = x^2 \).
2. **Apply Horizontal Shift**:
The function \( y = (x - 3)^2 \) represents a horizontal shift of the parent function \( y = x^2 \) to the right by 3 units.
3. **Apply Vertical Shift**:
The function \( y = (x - 3)^2 - 4 \) represents a vertical shift of the function \( y = (x - 3)^2 \) downward by 4 units.
### Graphing the Function:
1. **Draw the Coordinate Axes**:
Start by drawing the x-axis and y-axis with suitable scales. In the given image, the axes range from -5 to 5 on both scales.
2. **Plot the Vertex**:
The vertex of the function \( y = (x - 3)^2 - 4 \) is at the point (3, -4).
3. **Sketch the Parabola**:
From the vertex (3, -4), sketch the parabola opening upwards. Use symmetry about the vertex and plot additional points symmetrically on either side of the vertex for more accuracy.
### Explanation of the Given Graph:
The provided graph is a coordinate plane with the x-axis ranging from -5 to 5 and the y-axis ranging from -5 to 5. The graph is currently empty, ready for you to plot the given quadratic function using the transformations described above.
### Domain and Range:
**Domain**: The domain of \( y = (x - 3)^2 - 4 \) includes all real numbers, as the function is defined for all x-values. Thus, the domain is:
\[ \text{Domain: } (-\infty, \infty) \]
**Range**: The range of the function is determined by the vertex, which is the lowest point on the graph. Since the vertex is at (3, -4) and the parabola opens upwards, the y-values
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