Analyze x²-6x-5y=1. Determine the vertex, focus, directrix, intercepts, axis of symmetry, and at least 2 other points on the graph.
X^2 - 6X - 5Y = 1.
Solve for Y:
-5Y = -X^2 + 6X +1,
Divide both sides by -5:
Eq2: Y = X^2/5 - 6X/5 + 1/5,
h = Xv = -b/2a = (6/5) / (2/5) = 3.
Substitute 3 for X in Eq2:
k = Yv = 3^2/5 - 6*3/5 + 1/5,
k = 9/5 - 18/5 + 1/5,
k = 9/5 - 17/5 = - 8/5 = -1 3/5.
V(h , k) = V(3 , - 1 3/5 )
Axis = h = 3.
F(3 , Y2)
V(3 , - 1 3/5)
D(3 , Y1)
4a = 4(1/5) = 4/5.
1/4a = 5/4 = 1 1/4.
Y2 = K + 1/4a,
Y2 = - 8/5 + 5/4,
Common denominator = 20:
Y2 = - 32/20 + 25/20 = - 7/20.
Y1 = K - 1/4a,
Y1 = - 8/5 - 5/4,
Y1 = -8/5 - 5/4,
Common denominator = 20:
Y1 = -32/20 - 25/20,
Y1 = -57/20 = -2 17/20.
F(3 , -7/20)
V(3 , -1 3/5)
D(3 , -2 17/20)
P1(2 , -1 2/5)
P2(4 , -1 2/5)
X-Intercepts = 5.8 , 0.17.
Use Quadratic Formula.
CORRECTION: The constant in Eq2 should
be NEGATIVE 1/5.
Y = X^2/5 - 6X/5 - 1/5,
h = Xv = -b/2a = (6/5) / (2/5) = 3.
Substitute 3 for x in Eq2:
k=Yv=3^2/5 - 6*3/5 - 1/5 = - 10/5 =-2.
V(h , k) = V(3 , -2).
Axis = h = 3.
4a = 4(1/5 = 4/5).
1/4a = 5/4 = 1 1/4.
F(3 , Y2)
V(3 , -2)
D(3 , Y1)
Y2 = K + 1/4a = -2 + 5/4 = -3/4.
Y1=K-1/4a = -2 - 5/4 = -13/4 = -3 1/4.
F(3 , -3/4)
V(3 , -2)
D(3 , -3 3/4)
P1(2 , -1 4/5)
P2(4 , -1 4/5)
X-Intercepts = 6.16 and -0.16.
To analyze the equation x² - 6x - 5y = 1, we'll need to rewrite it in the standard form of a parabola equation, which is (x - h)² = 4p(y - k).
Step 1: Rewrite the equation in standard form:
x² - 6x - 5y = 1
Move the constant term to the other side:
x² - 6x - 1 = 5y
Divide the equation by 5 to simplify the equation further:
1/5(x² - 6x) = y
Step 2: Complete the square to determine the vertex.
To complete the square for the equation x² - 6x, take half of the coefficient of x (-6) and square it: (-6/2)² = 9.
Add 9 inside the parentheses and subtract 9 outside the parentheses to keep the equation balanced:
1/5(x² - 6x + 9 - 9) = y
Simplify the equation:
1/5((x - 3)² - 9) = y
Multiply both sides by 5 to eliminate the fraction:
(x - 3)² - 45 = 5y
Rearrange the equation in standard form:
(x - 3)² = 5y + 45
To compare the equation with (x - h)² = 4p(y - k), we have:
(x - 3)² = 5(y + 9)
The vertex of the parabola is therefore (h, k) = (3, -9).
Step 3: Determine the axis of symmetry.
The axis of symmetry is a vertical line passing through the vertex. In this case, the equation x = 3 represents the axis of symmetry.
Step 4: Determine the focus and directrix.
We can find the focus and directrix using the formula:
p = 1/4a
Comparing our equation to the standard form (x - h)² = 4p(y - k), we have a = 5.
Therefore, p = 1/(4 * 5) = 1/20.
The focus is located at (h, k + p) = (3, -9 + 1/20) = (3, -179/20).
The directrix is a horizontal line located at y = k - p, which is y = -9 - 1/20 or y = -181/20.
Step 5: Find the x-intercepts.
To find the x-intercepts, set y = 0 and solve for x.
x² - 6x - 1 = 0
Using the quadratic formula:
x = [ -b ± √(b² - 4ac) ] / 2a
a = 1, b = -6, c = -1
x = [ -(-6) ± √((-6)² - 4(1)(-1)) ] / 2(1)
x = (6 ± √(36 + 4)) / 2
x = (6 ± √40) / 2
x = (6 ± 2√10) / 2
x = 3 ± √10
Therefore, the x-intercepts are (3 + √10, 0) and (3 - √10, 0).
Step 6: Find at least two other points on the graph.
We can choose any x-values and substitute them into the equation to find the corresponding y-values.
Let's choose x = 1:
y = 1/5(1² - 6(1)) = -1
Therefore, one point on the graph is (1, -1).
Let's choose x = 5:
y = 1/5(5² - 6(5)) = -17
Therefore, another point on the graph is (5, -17).
In summary:
- Vertex: (3, -9)
- Focus: (3, -179/20)
- Directrix: y = -181/20
- X-intercepts: (3 + √10, 0) and (3 - √10, 0)
- Axis of symmetry: x = 3
- Other points on the graph: (1, -1) and (5, -17)
To analyze the equation x²-6x-5y=1, we need to first rewrite it in the standard form of a quadratic equation.
Step 1: Move the constant term to the right side of the equation:
x²-6x-5y-1=0
Step 2: Reorder the equation to group x-terms and y-terms:
x²-6x = 5y+1
Step 3: Divide the entire equation by the coefficient of x² (which is 1):
(x²-6x)/1 = (5y+1)/1
Step 4: Complete the square to express the left side as a perfect square trinomial:
(x²-6x+9) = (5y+1)/1 + 9
Step 5: Simplify the right side:
(x²-6x+9) = 5y + 1 + 9
Step 6: Combine like terms on the right side:
(x²-6x+9) = 5y + 10
Step 7: Rewrite the left side as a perfect square trinomial:
(x-3)² = 5y + 10
Now we can determine the key properties of the equation.
1. Vertex:
Comparing the equation to the standard form of a quadratic equation, we can see that the vertex is at (h, k), where h = 3 and k = -2 (since (x-3)² is equivalent to (x-h)²).
Therefore, the vertex is (3, -2).
2. Axis of Symmetry:
The axis of symmetry is a vertical line passing through the vertex. So, the equation of the axis of symmetry is x = 3.
3. Intercepts:
To find the x-intercepts, set y = 0 and solve for x:
(x-3)² = 10
Taking the square root of both sides:
x-3 = ±√(10)
x = 3 ± √(10)
So, the x-intercepts are (3 + √(10), 0) and (3 - √(10), 0).
To find the y-intercept, set x = 0 and solve for y:
(0-3)² = 5y + 10
9 = 5y + 10
5y = -1
y = -1/5
So, the y-intercept is (0, -1/5).
4. Focus and Directrix:
To determine the focus and directrix of the parabola, we need to find the value of p, which is the distance from the vertex to the focus or directrix.
In the standard form (x-h)² = 4p(y-k), the value of p is equal to 1/(4a), where a is the coefficient of y in the equation.
In this case, a = 5, so p = 1/(4*5) = 1/20.
The focus is located at (h, k + p), which is (3, -2 + 1/20) or (3, -39/20).
The directrix is a horizontal line at y = k - p, which is y = -2 - 1/20 or y = -41/20.
5. Additional Points on the Graph:
To find additional points on the graph, we can substitute some x-values into the equation and solve for y. For example:
When x = 4:
(4-3)² = 5y + 10
1 = 5y + 10
5y = -9
y = -9/5
So, another point on the graph is (4, -9/5).
Similarly, we can substitute other values of x to find more points on the graph.
By following these steps, you can analyze the given quadratic equation, find its vertex, focus, directrix, intercepts, axis of symmetry, and other points on the graph.