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
y = x^2-13x+36
clearly the y-intercept is (0,36)
y = x^2 - 13x + (13/2)^2 + 36 - (13/2)^2
y = (x - 13/2)^2 - 25/4
So, the vertex is at (13/2,-25/4)
y = x^2-13x+36
y = (x-4)(x-9)
The x-intercepts are at (0,4),(0,9)
The coefficient of x^2 is positive, so it opens up.
(C) No idea what you mean by the size of a parabola. Stupid question.
x^2 = 4py
has its focus at y=p
So, for 4y = x^2, the focus is at (0,1/4)
We have shifted the parabola right by 13/2 and down by 25/4, so the focus is at (0+13/2, 1/4-25/4) = (13/2,-6)
http://www.wolframalpha.com/input/?i=x^2%E2%88%9213x%2B36%E2%89%A40
To understand and answer the given quadratic inequality x^2-13x+36≤0, we need to follow a step-by-step process:
A) Determine the type of graph:
To determine the type of the graph, we look at the quadratic term coefficient (x^2). In this case, the coefficient is positive (+1), indicating a U-shaped graph or a parabola that opens upwards.
B) Determine if it goes up or down:
Since the coefficient of the x^2 term is positive, the graph goes upward.
C) Determine the size:
To determine the size, we can inspect the discriminant of the inequality. The discriminant can be found using the quadratic formula: discriminant = b^2 - 4ac. In this case, a = 1, b = -13, and c = 36. Substituting these values into the formula, we get: discriminant = (-13)^2 - 4(1)(36) = 169 - 144 = 25. Since the discriminant is positive (25 > 0), the quadratic inequality has two real solutions, which means the graph is neither stretched nor compressed.
D) Graph the equation:
To graph the quadratic inequality x^2-13x+36≤0, we start by finding the x-intercepts and marking them on a graph. The x-intercepts are the values of x where the quadratic inequality equals zero. To find them, we set the quadratic inequality equal to zero: x^2 - 13x + 36 = 0. This equation can be factored as (x - 4)(x - 9) = 0, which means the x-intercepts are x = 4 and x = 9. On a graph, we plot these points.
E) Define the vertex:
The vertex of a quadratic function or inequality can be found using the formula x = -b/(2a) and substituting this x-value into the function/inequality to find the corresponding y-value. In this case, a = 1 and b = -13. Substituting these values into the formula, we get x = -(-13)/(2*1) = 13/2 = 6.5. To find the y-coordinate, substitute x = 6.5 into the inequality: (6.5)^2 - 13(6.5) + 36 ≤ 0. Evaluating this inequality, we find that it is true. Therefore, the vertex is (6.5, y), where y is any value less than or equal to zero.
F) X-intercept:
The x-intercepts of the quadratic inequality represent the values of x where the graph intersects or crosses the x-axis. As mentioned earlier, the x-intercepts are x = 4 and x = 9.
G) Y-intercept:
The y-intercept of the quadratic inequality represents the value of y when x equals zero. To find it, we set x = 0 in the inequality: (0)^2 - 13(0) + 36 ≤ 0. Evaluating this inequality, we find it is false since 36 is greater than zero. Therefore, the quadratic inequality does not have a y-intercept.
H) Focal point:
A quadratic inequality does not have a focal point. Focal points are associated with parabolas that have a directrix and a focus when the equation is in the standard form.
In summary, to analyze the quadratic inequality x^2-13x+36≤0:
A) The graph is a U-shaped graph or parabola opening upwards.
B) The graph goes upward.
C) The size is neither stretched nor compressed.
D) The equation can be graphed, with x-intercepts at x = 4 and x = 9.
E) The vertex is at (6.5, y), where y is less than or equal to zero.
F) The x-intercepts are x = 4 and x = 9.
G) There is no y-intercept.
H) There is no focal point.