How do you find the x-intercept of a cubic graph? For example y=2x^3+2, also, how does the 2 in front of the x influence the graph when compared to having a one in front?
x-intercept is a point on the curve where it crosses the x-axis, thus y = 0. substituting,
y = 2x^3 + 2
0 = 2x^3 + 2
0 = x^3 + 1
0 = (x + 1)(x^2 - x + 1)
thus the only real solution is
x = -1 ; and the x-intercept is at
(-1, 0)
for the second question, when the numerical coefficient of x^3 increases, the graph becomes narrower.
hope this helps :)
Thanks, that was really helpful :)
To find the x-intercept of a cubic graph, you need to set the y-value equal to zero and solve for x. In the given equation, y = 2x^3 + 2, you want to find when y is equal to zero.
Setting y = 0:
0 = 2x^3 + 2
To isolate x, subtract 2 from both sides:
-2 = 2x^3
Now divide both sides by 2:
-1 = x^3
To solve for x, we need to take the cube root of both sides:
∛(-1) = ∛(x^3)
-1 = x
So the x-intercept of the given cubic graph y = 2x^3 + 2 is x = -1.
Now, let's discuss the influence of the coefficient in front of x (2 in this case) on the graph. In a general cubic function of the form y = ax^3 + b, the coefficient a determines whether the graph opens upwards or downwards.
If a is positive, like in y = 2x^3 + 2, the graph will open upwards. This means that as x moves to positive or negative infinity, y will also increase or decrease without bound, respectively. It results in a U-shaped graph.
If a is negative, e.g., y = -2x^3 + 2, the graph will open downwards. As x progresses to positive or negative infinity, y will approach negative or positive infinity, respectively. Hence, the graph will be an upside-down U-shape.
In summary, the coefficient in front of x affects the direction in which the cubic graph opens, either upwards or downwards.