take the indicated quadratic inequality x^2−13x+36≤0
A)determine the type of graph
B)does it go up or down
C) the size
D)graph the equation
E)define the vertex
F)X -intercept
G) Y-intercept
H)focal point
To answer the questions related to the quadratic inequality x^2−13x+36≤0, we will need to analyze its graph and characteristics.
A) Determine the type of graph:
To determine the type of graph, we need to examine the leading coefficient (coefficient of x^2). In this case, the leading coefficient is 1. Since the leading coefficient is positive, the graph will open upwards.
B) Does it go up or down:
Since the graph opens upwards, it goes up.
C) The size:
To determine the size of the graph, we need to analyze the solutions to the inequality x^2−13x+36≤0. We'll solve this quadratic inequality by finding the x-intercepts or roots.
First, we can factor the quadratic expression: x^2−13x+36=(x−4)(x−9).
Setting each factor to zero, we get:
x−4=0 ---> x=4
x−9=0 ---> x=9
The solutions to the inequality are x≤4 and x≥9. This indicates that the graph exists in the range x ∈ [4, 9]. Therefore, the size of the graph covers the interval [4, 9].
D) Graph the equation:
To graph the equation x^2−13x+36, plot the vertex, x-intercepts, and y-intercept (which we'll find in subsequent steps).
E) Define the vertex:
The vertex of the quadratic function can be determined using the formula x = -b/2a. For our quadratic equation x^2−13x+36, the coefficient of x^2 is 1 (a) and the coefficient of x is -13 (b).
Using x = -b/2a, we find:
x = -(-13)/(2*1) = 13/2 = 6.5
So the x-coordinate of the vertex is 6.5. To find the y-coordinate, substitute the x-coordinate back into the equation:
y = (6.5)^2 - 13(6.5) + 36 = 6.25
Therefore, the vertex of the equation is (6.5, 6.25).
F) X-intercept:
The x-intercepts are the points where the graph intersects the x-axis. In this case, we found the x-intercepts earlier when solving the inequality: x=4 and x=9.
G) Y-intercept:
The y-intercept is the point where the graph intersects the y-axis. To find the y-intercept, substitute x=0 into the equation: y = (0)^2 - 13(0) + 36 = 36.
Therefore, the y-intercept is (0, 36).
H) Focal point:
Quadratic inequalities do not have focal points like quadratic equations. Focal points are associated with the focus of a parabola, which occurs in quadratic equations rather than inequalities. The given quadratic inequality does not contain a squared term that would result in a parabola with a focal point.
In summary:
A) The graph is a parabola.
B) It goes up.
C) The size of the graph covers the x-values from 4 to 9.
D) Graph the equation using the information obtained.
E) The vertex is (6.5, 6.25).
F) The x-intercepts are 4 and 9.
G) The y-intercept is (0, 36).
H) The quadratic inequality does not have a focal point.