Question
The graph is Quadratic #1. The equation is Quadratic #2.
Compare the characteristics of these two quadratics. Select all characteristics that are the exact same for both quadratics.
Quadratic #1
graph shows a line that passes through (0,5) and then curves and ends at (2,9) then goes down and passes through (5,0)
Quadratic #2
y=x2−4x+5
(2 points)
Responses
The y-intercept is the same for both quadratics.
The y-intercept is the same for both quadratics.
The axis of symmetry is the same for both quadratics.
The axis of symmetry is the same for both quadratics.
The end behavior is the same for both quadratics.
The end behavior is the same for both quadratics.
The number of solutions is the same for both quadratics.
Compare the characteristics of these two quadratics. Select all characteristics that are the exact same for both quadratics.
Quadratic #1
graph shows a line that passes through (0,5) and then curves and ends at (2,9) then goes down and passes through (5,0)
Quadratic #2
y=x2−4x+5
(2 points)
Responses
The y-intercept is the same for both quadratics.
The y-intercept is the same for both quadratics.
The axis of symmetry is the same for both quadratics.
The axis of symmetry is the same for both quadratics.
The end behavior is the same for both quadratics.
The end behavior is the same for both quadratics.
The number of solutions is the same for both quadratics.
Answers
Answered by
GPT-4o mini
To compare the characteristics of the two quadratics, we'll analyze both Quadratic #1 described by the graph and Quadratic #2 given by the equation \( y = x^2 - 4x + 5 \).
1. **Y-intercept**:
- For Quadratic #2, we can find the y-intercept by evaluating the equation at \( x = 0 \):
\[
y = 0^2 - 4(0) + 5 = 5.
\]
- From the description of Quadratic #1, the graph passes through (0, 5), which means it also has a y-intercept of 5.
- **Conclusion**: The y-intercept is the same for both quadratics.
2. **Axis of symmetry**:
- The axis of symmetry for a quadratic expressed in the form \( y = ax^2 + bx + c \) can be calculated using the formula \( x = -\frac{b}{2a} \).
- For Quadratic #2 (\( a = 1, b = -4, c = 5 \)):
\[
x = -\frac{-4}{2 \cdot 1} = \frac{4}{2} = 2.
\]
- From the description of Quadratic #1, we can assume this quadratic is symmetric and based on the vertex's x-value (which we can infer is at x = 2 based on the described rise and fall).
- **Conclusion**: The axis of symmetry is the same for both quadratics.
3. **End behavior**:
- The end behavior of a quadratic function refers to what happens to \( y \) as \( x \) approaches positive or negative infinity. For both quadratics given that they open upwards (as Quadratic #2 has a positive leading coefficient), we can infer a similar behavior (as \( y \) goes to infinity in both cases).
- **Conclusion**: The end behavior is the same for both quadratics.
4. **Number of solutions**:
- The number of solutions can consist of real x-intercepts (roots).
- For Quadratic #2, we can find the roots using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1} = \frac{4 \pm \sqrt{16 - 20}}{2} = \frac{4 \pm \sqrt{-4}}{2}.
\]
- Since the discriminant (\( b^2 - 4ac \)) is negative, Quadratic #2 has no real roots.
- For Quadratic #1, if it ends at (5, 0) and also moves from above the x-axis to below, it indicates that there is likely one real solution, potentially at (5, 0).
- **Conclusion**: The number of solutions is **not** the same for both quadratics.
Based on the analysis, we find:
- **The y-intercept is the same for both quadratics.**
- **The axis of symmetry is the same for both quadratics.**
- **The end behavior is the same for both quadratics.**
The characteristic that is not the same is the number of solutions.
**Selected confirmations:**
1. The y-intercept is the same for both quadratics.
2. The axis of symmetry is the same for both quadratics.
3. The end behavior is the same for both quadratics.
**The number of solutions is not the same for both quadratics.**
1. **Y-intercept**:
- For Quadratic #2, we can find the y-intercept by evaluating the equation at \( x = 0 \):
\[
y = 0^2 - 4(0) + 5 = 5.
\]
- From the description of Quadratic #1, the graph passes through (0, 5), which means it also has a y-intercept of 5.
- **Conclusion**: The y-intercept is the same for both quadratics.
2. **Axis of symmetry**:
- The axis of symmetry for a quadratic expressed in the form \( y = ax^2 + bx + c \) can be calculated using the formula \( x = -\frac{b}{2a} \).
- For Quadratic #2 (\( a = 1, b = -4, c = 5 \)):
\[
x = -\frac{-4}{2 \cdot 1} = \frac{4}{2} = 2.
\]
- From the description of Quadratic #1, we can assume this quadratic is symmetric and based on the vertex's x-value (which we can infer is at x = 2 based on the described rise and fall).
- **Conclusion**: The axis of symmetry is the same for both quadratics.
3. **End behavior**:
- The end behavior of a quadratic function refers to what happens to \( y \) as \( x \) approaches positive or negative infinity. For both quadratics given that they open upwards (as Quadratic #2 has a positive leading coefficient), we can infer a similar behavior (as \( y \) goes to infinity in both cases).
- **Conclusion**: The end behavior is the same for both quadratics.
4. **Number of solutions**:
- The number of solutions can consist of real x-intercepts (roots).
- For Quadratic #2, we can find the roots using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1} = \frac{4 \pm \sqrt{16 - 20}}{2} = \frac{4 \pm \sqrt{-4}}{2}.
\]
- Since the discriminant (\( b^2 - 4ac \)) is negative, Quadratic #2 has no real roots.
- For Quadratic #1, if it ends at (5, 0) and also moves from above the x-axis to below, it indicates that there is likely one real solution, potentially at (5, 0).
- **Conclusion**: The number of solutions is **not** the same for both quadratics.
Based on the analysis, we find:
- **The y-intercept is the same for both quadratics.**
- **The axis of symmetry is the same for both quadratics.**
- **The end behavior is the same for both quadratics.**
The characteristic that is not the same is the number of solutions.
**Selected confirmations:**
1. The y-intercept is the same for both quadratics.
2. The axis of symmetry is the same for both quadratics.
3. The end behavior is the same for both quadratics.
**The number of solutions is not the same for both quadratics.**
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