Find the vertex, the line of symmetry, the maximum or minimum of the quadratic function, and graph the function.
f(x)=x^2-4x-3
What is the vertex? (Type an ordered pair)
What is the equation of the line of symmetry?
x=
What is the maximum/minimum of f(x)?
Is the value, f(2)= -7 a minimum or maximum?
F(x) = X^2 -4X -3.
V(h , k),
h = Xv = -b/2a = 4/2 = 2,
k = Yv = F(2) = 2^2 -4*2 -3 = -7,
V(2 , -7).
Line of sym.: X = h = 2.
F(2) = -7 = min. (Eq opens upward)
To find the vertex, line of symmetry, and maximum or minimum of a quadratic function, we can use the formula -b/(2a) to find the x-coordinate of the vertex, and then substitute this value into the original equation to find the y-coordinate of the vertex.
For the given quadratic function f(x) = x^2 - 4x - 3:
Step 1: Identify the coefficients a, b, and c from the quadratic function, where a is the coefficient of x^2, b is the coefficient of x, and c is the constant term.
In this case, a = 1, b = -4, and c = -3.
Step 2: Calculate the x-coordinate of the vertex using the formula -b/(2a).
x_vertex = -(-4)/(2*1) = 4/2 = 2.
Step 3: Substitute the x-coordinate of the vertex into the original equation to find the y-coordinate of the vertex.
f(2) = (2)^2 - 4(2) - 3 = 4 - 8 - 3 = -7.
So, the vertex is located at the coordinates (2, -7).
The equation of the line of symmetry is given by the x-coordinate of the vertex. Therefore, the equation of the line of symmetry is x = 2.
To determine whether the vertex represents the maximum or minimum of the function, we need to look at the coefficient of the x^2 term (a).
If a > 0, the parabola opens upwards, indicating that the vertex corresponds to a minimum point.
If a < 0, the parabola opens downwards, indicating that the vertex corresponds to a maximum point.
In this case, a = 1, which is greater than 0, so the vertex represents a minimum point.
Finally, f(2) = -7 represents the y-coordinate of the vertex. Since the vertex represents the minimum point, the value f(2) = -7 is the minimum value of the function.
To graph the function, plot the vertex at (2, -7) and choose a few x-values to calculate the corresponding y-values. Connect the plotted points to form a curve.