Given the following function, find:
(a) vertex
(b) axis of symmetry
(c) intercepts,
(d) domain
(e) range
(f) intervals where the function is increasing,
(g) intervals where the function is decreasing
Please show all of your work
f(x)=-7x^2-14x-3
you have a parabola opening downward
f(x) = -7(x+1)^2 + 4
That should help you answer the questions. If not, come on back and say where you get stuck.
To find the vertex, axis of symmetry, intercepts, domain, range, intervals of increasing and decreasing, here's what you need to do step by step:
(a) Finding the vertex:
The vertex of a quadratic function in the form f(x) = ax^2 + bx + c can be found using the formula x = -b/2a. In this case, the function is f(x) = -7x^2 - 14x - 3.
Using the formula, we can find x = -(-14)/(2*(-7)) = 14/(-14) = -1.
Now substitute this x-value back into the original function to find the y-coordinate: f(-1) = -7(-1)^2 - 14(-1) - 3 = -7 + 14 - 3 = 4.
Hence, the vertex is (-1, 4).
(b) Finding the axis of symmetry:
The axis of symmetry is the line that passes through the vertex and divides the parabola into two congruent halves. It is given by the equation x = h, where (h, k) is the vertex. In this case, the axis of symmetry is x = -1.
(c) Finding the intercepts:
To find the x-intercepts, we need to set f(x) equal to zero and solve for x.
Setting -7x^2 - 14x - 3 = 0 and solving the quadratic equation gives us x = -0.4286 and x = -0.0714. So the x-intercepts are approximately -0.4286 and -0.0714.
To find the y-intercept, we evaluate f(0):
f(0) = -7(0)^2 - 14(0) - 3 = -3. Hence, the y-intercept is (0, -3).
(d) Finding the domain:
The domain of a quadratic function is all real numbers. Hence, the domain of f(x) = -7x^2 - 14x - 3 is (-∞, ∞).
(e) Finding the range:
Since the coefficient of x^2 term is negative (-7), the parabola opens downward. Therefore, the range is (-∞, k], where k is the y-coordinate of the vertex. In this case, the range is (-∞, 4].
(f) Finding intervals of increasing:
The function is increasing when the slope of the line tangent to the graph is positive. In a quadratic function, this occurs in the interval where the coefficient of the x^2 term is positive. In this case, the coefficient is negative (-7), so the function is decreasing on the entire domain (-∞, ∞).
(g) Finding intervals of decreasing:
Similarly, the function is decreasing when the slope of the line tangent to the graph is negative. In a quadratic function, this occurs in the interval where the coefficient of the x^2 term is negative. In this case, the coefficient is negative (-7), so the function is decreasing on the entire domain (-∞, ∞).
In summary:
(a) Vertex: (-1, 4)
(b) Axis of symmetry: x = -1
(c) Intercepts: x-intercepts (-0.4286, 0) and (-0.0714, 0); y-intercept (0, -3)
(d) Domain: (-∞, ∞)
(e) Range: (-∞, 4]
(f) Intervals of increasing: None
(g) Intervals of decreasing: (-∞, ∞)