The light pattern from a fog light at the airport can be modeled by the equation x^2/4 - y^2/16=1. One of the points on the graph is at (3, 4.5) and one of the x-intercepts is -2. Find the coordinates of the three addditional points on the graph and the other x-intercept.
I'm pretty sure I have to check the line symmetry of the x and y-axis and the lines y=x and y=-x. But as for getting the other x-intercept, I have no idea what to do at all. Thanks in advance.
To get the x intercept for any equation let y=0 and solve.
You should recognize your equation to be that of a standard hyperbola, with symmetrical x and y intercepts.
so the other x intercept would be x=-2
the other 3 points would be (3,-4.5), (-3,4.5) and (-3,-4.5)
BTW, the y value is approximately 4.5, actually it would be √20
To find the coordinates of the additional points on the graph and the other x-intercept, let's first analyze the given equation and then follow a step-by-step approach.
The equation of the light pattern is given as x^2/4 - y^2/16 = 1. We can recognize this as a hyperbola, specifically a vertical hyperbola, because the term with x^2 is positive and the term with y^2 is negative.
Step 1: Identify the center
The center of the hyperbola can be derived from the equation by isolating the constants on one side:
x^2/4 - y^2/16 = 1
Rewriting the equation in the standard form:
(x - h)^2/a^2 - (y - k)^2/b^2 = 1
Comparing this with the given equation, we can determine that the center of the hyperbola is at (h, k), which in our case is (0, 0).
Step 2: Identify the vertices
The vertices of the hyperbola can be found using the center and the values of a and b. Since this is a vertical hyperbola, a represents the distance from the center to the vertices along the x-axis, and b represents the distance from the center to the vertices along the y-axis. In this case, a^2 = 4, so a = 2, and b^2 = 16, so b = 4. Using this information, we can determine that the vertices are located at (h, k ± b), which results in (0, ±4).
Step 3: Find the x-intercepts
To find the x-intercepts, we set y = 0 and solve for x. Substituting y = 0 into the equation, we have:
x^2/4 - 0^2/16 = 1
x^2/4 = 1
x^2 = 4
Taking the square root of both sides, we get:
x = ±2
So, the x-intercepts are -2 and 2.
Step 4: Find the additional points on the graph
To find the additional points on the graph, we can select values for x and solve for y using the given equation. Let's say we choose x = 1 and x = -1.
For x = 1:
y is found by substituting x = 1 into the equation:
(1)^2/4 - y^2/16 = 1
1/4 - y^2/16 = 1
y^2/16 = 1 - 1/4
y^2/16 = 3/4
y^2 = (3/4) * 16
y^2 = 12
Taking the square root of both sides, we obtain:
y = ± √12
Simplifying, we get:
y ≈ ± 3.464
So, the additional point on the graph with x = 1 is approximately (1, ± 3.464).
Similarly, for x = -1:
y is found by substituting x = -1 into the equation:
(-1)^2/4 - y^2/16 = 1
1/4 - y^2/16 = 1
y^2/16 = 1 - 1/4
y^2/16 = 3/4
y^2 = (3/4) * 16
y^2 = 12
Taking the square root of both sides, we obtain:
y = ± √12
Simplifying, we get:
y ≈ ± 3.464
So, the additional point on the graph with x = -1 is approximately (-1, ± 3.464).
In summary, the coordinates of the three additional points on the graph are approximately: (1, ± 3.464), (-1, ± 3.464), and the other x-intercept is (2, 0).