y^2-x-4=0
a)find all x and y intercepts of the graph implied by this equation. (use an appropriate equation).
b) test the graph for symmetry using algebra. (explain reasoning through algebra)
y^2 = x+4
let x = 0 ---> y^2 = 4 , y = ± 2
let y = 0
x+4=0 ----> x = -4
A quick sketch will show a parabola with vertex (-4,0), opening to the right, and axis of symmetry the x-axis
To find the x-intercepts of the graph implied by the equation y^2 - x - 4 = 0, we can set y = 0 and solve for x.
Setting y = 0, we have 0^2 - x - 4 = 0, which simplifies to -x - 4 = 0. Adding x on both sides gives us x = -4. So, the x-intercept is (-4, 0).
To find the y-intercepts, we can set x = 0 and solve for y.
Setting x = 0, we have y^2 - 0 - 4 = 0, which simplifies to y^2 - 4 = 0. Rearranging the equation, we have y^2 = 4. Taking the square root of both sides, we get y = ±2. So, the y-intercepts are (0, 2) and (0, -2).
Now let's test the graph for symmetry using algebra.
a) Testing for symmetry with respect to the y-axis:
If a graph is symmetrical with respect to the y-axis, it means that replacing x with -x in the equation should yield the same equation.
Substituting -x for x in the given equation, we have y^2 - (-x) - 4 = 0, which simplifies to y^2 + x - 4 = 0.
Comparing this with the original equation, y^2 - x - 4 = 0, we notice that the two equations are not the same. Therefore, the graph is not symmetrical with respect to the y-axis.
b) Testing for symmetry with respect to the x-axis:
If a graph is symmetrical with respect to the x-axis, it means that replacing y with -y in the equation should yield the same equation.
Substituting -y for y in the given equation, we have (-y)^2 - x - 4 = 0, which simplifies to y^2 - x - 4 = 0.
Comparing this with the original equation, y^2 - x - 4 = 0, we notice that the two equations are the same. Therefore, the graph is symmetrical with respect to the x-axis.
In summary, the x-intercept is (-4, 0), the y-intercepts are (0, 2) and (0, -2), the graph is not symmetrical with respect to the y-axis, but it is symmetrical with respect to the x-axis.