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 Quadratic #2 y=x2−4x+5 (2 points) Responses The number of solutions is the same for both quadratics. The number of solutions 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 y-intercept is the same for both quadratics. The y-intercept is the same for both quadratics. The end behavior is the same for both quadratics. The end behavior is the same for both quadratics. Skip to navigation
Answers
Answered by
GPT-4o mini
To compare the characteristics of the two quadratics based on the information given, let's analyze the first quadratic equation provided:
**Quadratic #1:** \(y = x^2 - 4x + 5\)
1. **Number of solutions:** The number of solutions (roots) can be determined by calculating the discriminant \(D\) from the quadratic formula \(D = b^2 - 4ac\). Here, \(a = 1\), \(b = -4\), and \(c = 5\):
\[
D = (-4)^2 - 4(1)(5) = 16 - 20 = -4
\]
Since the discriminant is negative, Quadratic #1 has no real solutions.
2. **Axis of symmetry:** The axis of symmetry can be found using the formula \(x = -\frac{b}{2a}\):
\[
x = -\frac{-4}{2(1)} = \frac{4}{2} = 2
\]
So the axis of symmetry for Quadratic #1 is \(x = 2\).
3. **Y-intercept:** The y-intercept can be found by evaluating the function when \(x = 0\):
\[
y = 0^2 - 4(0) + 5 = 5
\]
Therefore, the y-intercept is \(5\).
4. **End behavior:** As \(x\) approaches positive or negative infinity, the quadratic term \(x^2\) dominates. Thus, the end behavior is the same for all quadratic functions (both ends go toward positive infinity).
Now, for **Quadratic #2**, we do not have any information provided about its equation, so we cannot compare it directly. However, the characteristics that you can compare between both quadratics (once you determine the equation of Quadratic #2) will be:
- The number of solutions
- Axis of symmetry
- Y-intercept
- End behavior
If Quadratic #2 has the same discriminant, axis of symmetry, y-intercept, and end behavior as Quadratic #1, then the characteristics you select will be the same.
You should check Quadratic #2 for these properties to complete the comparisons. If you find Quadratic #2 has the same properties as Quadratic #1, then you can select those characteristics accordingly.
**Quadratic #1:** \(y = x^2 - 4x + 5\)
1. **Number of solutions:** The number of solutions (roots) can be determined by calculating the discriminant \(D\) from the quadratic formula \(D = b^2 - 4ac\). Here, \(a = 1\), \(b = -4\), and \(c = 5\):
\[
D = (-4)^2 - 4(1)(5) = 16 - 20 = -4
\]
Since the discriminant is negative, Quadratic #1 has no real solutions.
2. **Axis of symmetry:** The axis of symmetry can be found using the formula \(x = -\frac{b}{2a}\):
\[
x = -\frac{-4}{2(1)} = \frac{4}{2} = 2
\]
So the axis of symmetry for Quadratic #1 is \(x = 2\).
3. **Y-intercept:** The y-intercept can be found by evaluating the function when \(x = 0\):
\[
y = 0^2 - 4(0) + 5 = 5
\]
Therefore, the y-intercept is \(5\).
4. **End behavior:** As \(x\) approaches positive or negative infinity, the quadratic term \(x^2\) dominates. Thus, the end behavior is the same for all quadratic functions (both ends go toward positive infinity).
Now, for **Quadratic #2**, we do not have any information provided about its equation, so we cannot compare it directly. However, the characteristics that you can compare between both quadratics (once you determine the equation of Quadratic #2) will be:
- The number of solutions
- Axis of symmetry
- Y-intercept
- End behavior
If Quadratic #2 has the same discriminant, axis of symmetry, y-intercept, and end behavior as Quadratic #1, then the characteristics you select will be the same.
You should check Quadratic #2 for these properties to complete the comparisons. If you find Quadratic #2 has the same properties as Quadratic #1, then you can select those characteristics accordingly.
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