For the quadratic function f(x) = 2x^2 - 16x + 27
The Vertex = ___?
f(x) attains a maximum minimum value of ___?
The axis of symmetry is___?
The x-intercepts are___? and The y-intercept is ____?
x of the vertex is -b/(2a) -----> good formula to memorize
= -(-16)/(4) = 4
f(4) = ....
so now you have the vertex.
easy to read off the max/min and the axis of symmetry.
for x intercept let y = 0 in y = 2x^2 - 16x + 27
for y intercept let x = 0 in y = 2x^2 - 16x + 27
confirm my looking at the graph:
http://www.wolframalpha.com/input/?i=y+%3D+2x%5E2+-+16x+%2B+27
To find the vertex of a quadratic function in the form f(x) = ax^2 + bx + c, we can use the formula:
x = -b / (2a)
For the given quadratic function f(x) = 2x^2 - 16x + 27, let's apply the formula to find the vertex:
a = 2, b = -16
x = (-(-16)) / (2 * 2)
x = 16 / 4
x = 4
To find the y-coordinate of the vertex, substitute the x-value back into the original function:
f(x) = 2(4)^2 - 16(4) + 27
f(x) = 2(16) - 64 + 27
f(x) = 32 - 64 + 27
f(x) = -5
Therefore, the vertex of the given quadratic function is (4, -5).
To determine if the parabola opens upward or downward, we can look at the coefficient of the x^2 term (a). If a > 0, the parabola opens upward, and if a < 0, it opens downward. In this case, a = 2, so the parabola opens upward.
Since the parabola opens upward and the vertex is the lowest point on the graph, the function has a minimum value at the vertex. Therefore, the function attains a minimum value of -5.
The axis of symmetry is a vertical line that passes through the vertex of a parabola. In a quadratic function of the form f(x) = ax^2 + bx + c, the axis of symmetry is given by the equation:
x = -b / (2a)
Using the values a = 2 and b = -16, we can find the axis of symmetry:
x = -(-16) / (2 * 2)
x = 16 / 4
x = 4
Therefore, the axis of symmetry is the vertical line x = 4.
To find the x-intercepts (also known as roots or zeros) of the quadratic function, we need to solve the equation f(x) = 0. In this case, we have the equation:
2x^2 - 16x + 27 = 0
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. However, in this case, the equation does not factor easily, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the coefficients a = 2, b = -16, and c = 27 into the quadratic formula, we get:
x = (-(-16) ± √((-16)^2 - 4 * 2 * 27)) / (2 * 2)
x = (16 ± √(256 - 216)) / 4
x = (16 ± √40) / 4
Simplifying further:
x = (16 ± 2√10) / 4
x = 4 ± 0.5√10
Therefore, the x-intercepts are approximately:
x ≈ 4 + 0.5√10
x ≈ 4 - 0.5√10
Finally, to find the y-intercept, we substitute x = 0 into the original function:
f(x) = 2(0)^2 - 16(0) + 27
f(x) = 27
Therefore, the y-intercept is (0, 27).