Given the following quadratic equation, find
a. the vertex
b. the axis of symmetry
c. the intercepts
d. the domain
e. the range
f. the interval where the function is increasing, and
g. the interval where the function is decreasing
h. Graph the function.
y=x^2+4x
The answer to this question is 0.
change it to the standard form
y = a(x-p)^2 + q , by completing the square
y = x^2 = 4x
= x^2 + 4x + 4 - 4
= (x + 2)^2 - 4
All the above can now be answered readily.
The form y = a(x-p)^2 + q
should be in your text or in your notes from class.
To find the information about the given quadratic equation, let's go step by step:
a. To find the vertex of the quadratic equation y = x^2 + 4x, we can use the formula (-b/2a, f(-b/2a)), where a, b, and c are the coefficients of the quadratic equation in the form ax^2 + bx + c. In this case, the equation is already in the form y = x^2 + 4x, so a = 1 and b = 4. Plugging these values into the formula, we get:
x-coordinate of vertex = -b/2a = -4/(2*1) = -2
Now, to find the y-coordinate of the vertex, we substitute the x-coordinate (-2) back into the equation:
y = (-2)^2 + 4(-2) = 4 - 8 = -4
Therefore, the vertex of the quadratic equation is (-2, -4).
b. The axis of symmetry is a vertical line that passes through the vertex. Its equation can be found by simply using the x-coordinate of the vertex. In this case, the equation of the axis of symmetry is x = -2.
c. To find the intercepts, we need to find the points at which the quadratic equation intersects the x and y-axes.
For x-intercepts, we set y = 0 and solve for x. So, we have:
0 = x^2 + 4x
Factoring out x, we get:
0 = x(x + 4)
Setting each factor equal to zero, we find:
x = 0 or x + 4 = 0
Solving for x, we get:
x = 0 or x = -4
Therefore, the x-intercepts are (0, 0) and (-4, 0).
For the y-intercept, we set x = 0 and find the corresponding y-value:
y = 0^2 + 4(0) = 0
Therefore, the y-intercept is (0, 0).
d. The domain of a quadratic function is all real numbers, so the domain is (-∞, ∞).
e. To determine the range, we consider the vertex. Since the coefficient of x^2 (a) is positive in this case, the parabola opens upward, and the vertex represents the minimum point. Thus, the range is all y-values greater than or equal to the y-coordinate of the vertex. In this case, the range is [-4, ∞).
f. To find where the function is increasing, we look at the graph. Since the coefficient of x^2 is positive, the parabola opens upward. Therefore, the function is increasing on the interval (-∞, -2).
g. Similarly, since the parabola opens upward, the function is decreasing on the interval (-2, ∞).
h. Graphing the function, we plot the vertex (-2, -4), the x-intercepts (0, 0) and (-4, 0), and the y-intercept (0, 0). Drawing a smooth curve passing through these points, we get a parabola opening upward.
Here is the graph of the function y = x^2 + 4x:
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