Find the x and y intercepts for the following parabolas.
y=(x+12)^2-144
y=(x-8)^-4
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To find the x-intercepts, we set y equal to 0 and solve for x.
For the first parabola, y = (x+12)^2 - 144:
Setting y equal to 0:
0 = (x+12)^2 - 144
Next, we can add 144 to both sides of the equation:
144 = (x+12)^2
Then, taking the square root of both sides:
± 12 = x+12
Finally, subtracting 12 from both sides:
± 12 - 12 = x
x = ± 0
Therefore, the x-intercepts for the first parabola are (0, 0).
For the second parabola, y = (x-8)^-4:
Setting y equal to 0:
0 = (x-8)^-4
Since the exponent is negative, we take the reciprocal of both sides:
1/0 = (x-8)^4
Since anything raised to the power of 4 is never undefined, there is no x-intercept for this second parabola.
Moving on to finding the y-intercept, we set x equal to 0 and solve for y.
For the first parabola, y = (x+12)^2 - 144:
Setting x equal to 0:
y = (0+12)^2 - 144
Simplifying:
y = 12^2 - 144 = 144 - 144 = 0
Therefore, the y-intercept for the first parabola is (0, 0).
For the second parabola, y = (x-8)^-4:
Setting x equal to 0:
y = (0-8)^-4
Simplifying:
y = (-8)^-4 = 1/(-8)^4 = 1/4096
Therefore, the y-intercept for the second parabola is (0, 1/4096).
To find the x and y intercepts of a parabola, you need to set one of the variables to zero and solve for the other variable.
Let's start with the first parabola: y = (x+12)^2 - 144
➊ X-intercept:
To find the x-intercept, we set y to zero and solve for x. So we have:
0 = (x+12)^2 - 144
Next, we can simplify the equation by adding 144 to both sides:
144 = (x+12)^2
Now, we take the square root of both sides:
±√144 = x + 12
Simplifying, we have:
x = -12 ± 12
So the x-intercepts are x = -24 and x = 0.
➋ Y-intercept:
To find the y-intercept, we set x to zero and solve for y. So we have:
y = (0+12)^2 - 144
y = 144 - 144
y = 0
Therefore, the y-intercept is y = 0.
Now let's move on to the second parabola: y = (x-8)^-4
➊ X-intercept:
Setting y to zero, we have:
0 = (x-8)^-4
To solve this equation, we need to determine the value of x such that (x-8) raised to the power of -4 equals zero. However, this equation does not have any solution since negative numbers cannot be raised to a negative exponent and result in zero.
➋ Y-intercept:
To find the y-intercept, we set x to zero and solve for y. So we have:
y = (0-8)^-4
y = (-8)^-4
y = 1/(-8)^4
y = 1/4096
Therefore, the y-intercept is y = 1/4096.
In summary:
The x-intercepts for the first parabola are x = -24 and x = 0, and the y-intercept is y = 0.
The second parabola does not have any x-intercepts, and the y-intercept is y = 1/4096.