analyze the graph of f(x)=(x+3)x-1)^2
x and y itercepts; end behavior; and maximum number of turning points
repost with balanced parentheses, so we know what we're working with.
sorry-operating on no sleep for at least 24 hours!
f(x)=(x+3)(x-1)^2
analyze x and y intercepts; end behavior and maximum number of turning points.
y-intercept at x=0: 3*1=3
x-intercepts: -3,1,1
since two roots at x=1, graph will be tangent there, not cross.
x gets large, y gets large; degree 3
degree 3, max turning points: 2
Thanks, Steve -
I cn never get my graphs right - I get all the work, but to draw?! forget it.
To analyze the graph of the function f(x) = (x + 3)(x - 1)^2, let's break down the steps to determine the x-intercepts, y-intercepts, end behavior, and maximum number of turning points.
1. X-intercepts:
X-intercepts occur where the graph of a function intersects the x-axis. To find them, we set f(x) = 0 and solve for x:
(x + 3)(x - 1)^2 = 0
Setting each factor equal to zero:
x + 3 = 0 or (x - 1)^2 = 0
Solving for x, we get two x-intercepts:
x + 3 = 0 --> x = -3 (x-intercept at x = -3)
(x - 1)^2 = 0 --> x = 1 (x-intercept at x = 1)
2. Y-intercept:
The y-intercept occurs where the graph of the function intersects the y-axis. To find it, we substitute x = 0 into the function:
f(0) = ((0 + 3)(0 - 1)^2)
f(0) = (3)(1) = 3
Therefore, the y-intercept is at y = 3.
3. End Behavior:
End behavior refers to how the function behaves as x approaches positive or negative infinity. To determine this, we look at the degree and leading coefficient of the polynomial.
In this case, the function f(x) = (x + 3)(x - 1)^2 is a polynomial of degree 3 with a positive leading coefficient. As x approaches positive or negative infinity, the function will also approach positive infinity.
4. Maximum Number of Turning Points:
The maximum number of turning points is equal to the highest degree value minus 1. In this case, the highest degree is 3, so the maximum number of turning points is 3 - 1 = 2.
To summarize:
- X-intercepts: x = -3 and x = 1
- Y-intercept: y = 3
- End Behavior: As x approaches positive or negative infinity, y approaches positive infinity
- Maximum Number of Turning Points: 2