use the eight step strategy to graph
f(x)=-3x2-12x-5
find the vertex
find x and y intercepts
determine the number of points to graph
graph the parabola
f(x) = -3(x^2+4x) - 5
= -3(x^2+4x+4) - 5 +3(4)
= -3(x+2)^2 + 7
so, the vertex is at (-2,7)
clearly, (0,-5) is the y-intercept
solve for x to get the two x-intercepts
see
http://www.wolframalpha.com/input/?i=-3x^2-12x-5
To graph the function f(x) = -3x^2 - 12x - 5 using the eight-step strategy, follow these steps:
Step 1: Rewrite the equation
f(x) = -3x^2 - 12x - 5
Step 2: Identify the coefficients
a = -3 (coefficient of x^2)
b = -12 (coefficient of x)
c = -5 (constant term)
Step 3: Determine the vertex
The x-coordinate of the vertex (h) can be found using the formula:
h = -b / (2a)
Substituting the values, we get:
h = -(-12) / (2 * -3)
h = 12 / -6
h = -2
To find the y-coordinate of the vertex (k), substitute the value of h into the equation:
k = f(-2)
k = -3 * (-2)^2 - 12 * (-2) - 5
k = -3 * 4 + 24 - 5
k = -12 + 24 - 5
k = 7
Therefore, the vertex is (-2, 7).
Step 4: Find the x-intercepts
To find the x-intercepts, set f(x) = 0 and solve for x. In this case, we solve the quadratic equation:
-3x^2 - 12x - 5 = 0
You can solve this equation by factoring, using the quadratic formula, or completing the square. In this example, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Substitute the values and solve for x:
x = (-(-12) ± √((-12)^2 - 4 * (-3) * (-5))) / (2 * (-3))
x = (12 ± √(144 - 60)) / (-6)
x = (12 ± √(84)) / (-6)
Simplifying further, we have:
x = (12 ± 2√21) / (-6)
x = (2 ± √21) / (-1)
x = -(2 ± √21)
Therefore, the x-intercepts are approximately x = - (2 + √21) and x = - (2 - √21).
Step 5: Find the y-intercept
To find the y-intercept, substitute x = 0 into the function:
f(0) = -3(0)^2 - 12(0) - 5
f(0) = -5
Therefore, the y-intercept is (0, -5).
Step 6: Determine the number of points to graph
Since the vertex is the highest or lowest point on the parabola (depending on the sign of 'a'), and in this case 'a' is negative (-3), the parabola opens downward. Therefore, there are no maximum points and an infinite number of points to graph.
Step 7: Plot the points
Using the information calculated above, plot the points on a graph:
- Vertex (-2,7)
- X-intercepts approximately x = - (2 + √21) and x = - (2 - √21)
- Y-intercept (0, -5)
Step 8: Graph the parabola
Connect the points with a smooth curve, making sure the parabola opens downward and is symmetric about the vertex (-2,7).