Question
On a piece of paper, graph this system of equations.
y =x-2
y=x2-5x+6
Then determine which answer choice matches the graph you drew and identify the solutions to the system.
y =x-2
y=x2-5x+6
Then determine which answer choice matches the graph you drew and identify the solutions to the system.
Answers
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Answered by
GPT-4o mini
To graph the system of equations \(y = x - 2\) and \(y = x^2 - 5x + 6\), let's first analyze each equation.
1. **Linear Equation: \(y = x - 2\)**:
- This is a straight line with a slope of 1 and a y-intercept of -2. It passes through the point (2, 0) and (0, -2).
2. **Quadratic Equation: \(y = x^2 - 5x + 6\)**:
- This is a parabola that opens upwards. We can find the roots (x-intercepts) by factoring:
\[
y = (x - 2)(x - 3)
\]
- The roots are \(x = 2\) and \(x = 3\), indicating the parabola intersects the x-axis at these points.
3. **Finding the vertex:**
- The vertex can be found using the formula \(x = -\frac{b}{2a}\) where \(a = 1\) and \(b = -5\):
\[
x = -\frac{-5}{2 \times 1} = \frac{5}{2} = 2.5
\]
- Substitute \(x = 2.5\) back into the equation for \(y\):
\[
y = (2.5)^2 - 5(2.5) + 6 = 6.25 - 12.5 + 6 = -0.25
\]
- So, the vertex of the parabola is at \((2.5, -0.25)\).
### Graphing the Functions:
- The linear function is a straight line.
- The quadratic function is a parabola with the vertex and intercept points calculated.
### Determining the Intersections:
To find the points at which these two functions intersect (the solutions to the system), we set the equations equal to each other:
\[
x - 2 = x^2 - 5x + 6
\]
Rearranging gives:
\[
0 = x^2 - 6x + 8
\]
Next, we factor the quadratic:
\[
0 = (x - 2)(x - 4)
\]
So, the solutions are:
\[
x = 2 \quad \text{and} \quad x = 4.
\]
### Finding y-values for the solutions:
- For \(x = 2\):
\[
y = 2 - 2 = 0
\]
Point: \((2, 0)\).
- For \(x = 4\):
\[
y = 4 - 2 = 2
\]
Point: \((4, 2)\).
### Conclusion:
The system of equations intersects at points \((2, 0)\) and \((4, 2)\).
To graph this, you'd see the line intersecting the parabola at these two points. If you have multiple choice answers, look for a graph that shows a linear line crossing a parabola at approximately those coordinates. The solutions to the system are:
\[
(2, 0) \quad \text{and} \quad (4, 2).
\]
1. **Linear Equation: \(y = x - 2\)**:
- This is a straight line with a slope of 1 and a y-intercept of -2. It passes through the point (2, 0) and (0, -2).
2. **Quadratic Equation: \(y = x^2 - 5x + 6\)**:
- This is a parabola that opens upwards. We can find the roots (x-intercepts) by factoring:
\[
y = (x - 2)(x - 3)
\]
- The roots are \(x = 2\) and \(x = 3\), indicating the parabola intersects the x-axis at these points.
3. **Finding the vertex:**
- The vertex can be found using the formula \(x = -\frac{b}{2a}\) where \(a = 1\) and \(b = -5\):
\[
x = -\frac{-5}{2 \times 1} = \frac{5}{2} = 2.5
\]
- Substitute \(x = 2.5\) back into the equation for \(y\):
\[
y = (2.5)^2 - 5(2.5) + 6 = 6.25 - 12.5 + 6 = -0.25
\]
- So, the vertex of the parabola is at \((2.5, -0.25)\).
### Graphing the Functions:
- The linear function is a straight line.
- The quadratic function is a parabola with the vertex and intercept points calculated.
### Determining the Intersections:
To find the points at which these two functions intersect (the solutions to the system), we set the equations equal to each other:
\[
x - 2 = x^2 - 5x + 6
\]
Rearranging gives:
\[
0 = x^2 - 6x + 8
\]
Next, we factor the quadratic:
\[
0 = (x - 2)(x - 4)
\]
So, the solutions are:
\[
x = 2 \quad \text{and} \quad x = 4.
\]
### Finding y-values for the solutions:
- For \(x = 2\):
\[
y = 2 - 2 = 0
\]
Point: \((2, 0)\).
- For \(x = 4\):
\[
y = 4 - 2 = 2
\]
Point: \((4, 2)\).
### Conclusion:
The system of equations intersects at points \((2, 0)\) and \((4, 2)\).
To graph this, you'd see the line intersecting the parabola at these two points. If you have multiple choice answers, look for a graph that shows a linear line crossing a parabola at approximately those coordinates. The solutions to the system are:
\[
(2, 0) \quad \text{and} \quad (4, 2).
\]
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