f(x)=-3x2-12x-5
find x intercepts
a. analyze the multiplicity at each intercept (where it crossed/touches the axis)
determine the number of turning points
find four more points to graph
graph the parabola
did you check your earlier post?
http://www.jiskha.com/display.cgi?id=1425519642
Was there something unclear there?
To find the x-intercepts of the function f(x) = -3x^2 - 12x - 5, we need to set the function equal to zero and solve for x.
1. Set the equation equal to zero: -3x^2 - 12x - 5 = 0.
Now, let's solve this quadratic equation by factoring, using the quadratic formula, or completing the square. In this case, it cannot be factored easily, so the quadratic formula will be used.
2. Apply the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
For our equation, a = -3, b = -12, and c = -5. Plugging these values into the formula, we get:
x = (-(-12) ± √((-12)^2 - 4(-3)(-5))) / (2(-3))
= (12 ± √(144 - 60)) / (-6)
= (12 ± √(84)) / (-6)
= (12 ± 2√(21)) / (-6)
3. Simplify the expression:
The two x-intercepts can be found by calculating two values:
x₁ = (12 + 2√(21)) / (-6)
x₂ = (12 - 2√(21)) / (-6)
Now, let's proceed to analyze the multiplicity at each intercept and determine the turning points.
4. Multiplicity:
Multiplicity indicates the number of times a particular intercept crosses or touches the x-axis.
In a quadratic equation like f(x) = -3x^2 - 12x - 5, the multiplicity of the intercept is always 2. This means it crosses the x-axis twice.
5. Turning Points:
To determine the number of turning points for a quadratic equation, we consider its leading coefficient, which is -3 in this case. Since the coefficient is negative, the parabola opens downwards, and there will be one turning point.
6. Finding additional points to graph:
Choose any value of x other than the intercepts and calculate f(x) to find corresponding points on the graph.
Let's choose x = -2, -1, 0, and 1.
For x = -2:
f(-2) = -3(-2)^2 - 12(-2) - 5
= -3(4) + 24 - 5
= -12 + 24 - 5
= 7
So, one of the additional points is (-2, 7).
Similarly, calculate the values for x = -1, 0, and 1 to find the other points on the graph.
7. Graphing the parabola:
Collect all the information to graph the parabola:
- Intercepts: (x₁, 0) and (x₂, 0)
- Multiplicity: 2
- Turning Points: 1
- Additional Points: (-2, 7), (-1, -6), (0, -5), (1, -9)
Now plot these points on a graph, and you will have the graph of the parabola.