Answer:
Vertex: (3, -1). x-intercept(s): (4,0) & (2,0). y-intercept: (0,8). Domain: (-∞,∞). Range: [-1,∞) Axis of symmetry: x= 3
Explanation:
the standard form of a parabola is ax² + bx + c which is also known as a quadratic equation
the vertex form of a parabola is a(x - h)² + k
problem 1: x² - 6x + 8
to solve this problem, we would need to put the equation into vertex form.
to find the vertex, we will use the formula -b/2a, where b is -6 and a is 1 to find h (also known as the x-coordinate). plug it in to the formula
-(-6)/2(1) = 6/2 = 3 <-- the x-coordinate
to find the y-coordinate, we will plug in the newly found x-coordinate into the original equation
solve: x² - 6x + 8 when x = 3
(3)² - 6(3) + 8 = 9 - 18 + 8 = -1 <-- the y-coordinate
so, the vertex of the equation is (3, -1)
next, we determine if the parabola opens up or down. in this case, the parabola opens up because the coefficient infront of x² is positive (1). if the coefficient was negative (-1), the parabola would open down.
now, we find the x & y intercepts. you can plot these coordinates in your table of values.
finding the x intercept(s): solve the equation for 0
solving x² - 6x + 8 =0, we can factor this out to be (x - 4)(x - 2) = 0. using the zero-product rule, we solve each of those equations for 0, with x - 4 = 0 and x - 2 = 0. the results from these are 4 and 2.
this makes the x intercepts (4, 0) and (2, 0)
finding the y intercept(s): substitute x = 0 in the original equation
substituting 0 for x in the original equation gives us:
(0)² - 6(0) + 8, which gives us only 8 since the 0s cancel out everything else. you can also say that the constant term c is the y intercept in order to make things a bit easier.
this makes the y intercept (0, 8)
finding the domain of a parabola:
the domain of a parabola is always (-∞,∞), because it continues from left to right indefinitely
finding the range of a parabola:
the range of a parabola goes from bottom to top. the lowest y coordinate that the parabola has is located at (3, -1), making the lowest y coodinate -1. since the parabola opens upward, we know that it continues to positive infinity.
the range of the parabola is [-1,∞). we use a bracket on -1 because the point is located on the graph and is included in the solution.
the axis of symmetry: the axis of symmetry is the x-coordinate of the vertex, which is 3. so, the axis of symmetry would be x=3.
to find additional points, plug in any real number into the original equation, and the output will be the y value. for example:
x² - 6x +8 when x = 1
(1)² - 6(1) + 8 -> 1 - 6 + 8= 3
so, another coordinate on the grid would be (1, 3).
with all of this information, you can graph the parabola starting at the vertex and going through the y & x intercept(s) and the oher points you may need on the graph, along with solving the other equation.
i have attached a roughly drawn parabola of what it should look similar to, listing the intercepts and the vertex.