Answer:
Graphs Attached Below
Step-by-step explanation:
Hello!
Standard form of a quadratic: [tex]ax^2 + bx + c= 0[/tex]
From our Equation:
There are several values that are needed to drawing a parabola:
- y - intercept
- Axis of Symmetry (AOS)
- Vertex
- x - intercepts
Y-intercept
Standard form of a quadratic: [tex]ax^2 + bx + c= 0[/tex]
The y-intercept is the "c" value. Given that our equation has a "c" value of 0, the y -intercept is 0.
Axis of Symmetry
A parabola is always symmetrical vertically. The line in which the fold happens is the Axis of Symmetry.
To calculate the AOS, we use the formula [tex]AOS = \frac{-b}{2a}[/tex], from the values of the equation.
Calculate
- [tex]AOS = \frac{-b}{2a}[/tex]
- [tex]AOS = \frac{-(-2)}{2(0.5)}[/tex]
- [tex]AOS = \frac{2}{1}[/tex]
- [tex]AOS = 2[/tex]
The Axis of Symmetry is a vertical line, so the AOS is the line x = 2.
Vertex
The vertex is the highest or lowest point on the graph of a parabola. It resides on the AOS of the graph.
To calculate the vertex, we simply have to find the y-value, given that we have the x-value from the AOS. We can find the y-value by plugging in the AOS for x in the original equation.
Calculate
- [tex]f(x) = \frac12x^2 - 2x[/tex]
- [tex]f(x) = \frac12 (2)^2 - 2(2)[/tex]
- [tex]f(x) = 2 - 4[/tex]
- [tex]f(x) = -2[/tex]
The y-value is -2. The vertex is (2, -2).
X-intercepts
The x-intercepts are the points where the graph intersects the x-axis (y = 0).
Solve by Factoring
- [tex]f(x) = \frac12 x^2 - 2x[/tex]
- [tex]0 = \frac12x(x - 4)[/tex]
- [tex]x = 0, x = 4[/tex]
The roots are (0,0) and (4,0).
Graph
Now we just draw the y-intercept, vertex, AOS, and the x-intercepts, and draw a curved line between them.
Image Attached
Domain Restrictions
The Domain (x-values) are being restricted to all x-values that are greater than or equal to -2 and less than 4.
That means we remove the parts of the line that don't belong in that domain.
Image Attached
