14) Consider the parabola with equation y = x^2 - 6x + 5.
a. Use any suitable method to determine the coordinates of the turning point of this parabola.
b. Hence, state for which values of c the line y = c will intersect the parabola:
i. twice
ii. once
iii. not at all
You are probably expected to use "completing the square", although other methods are just as suitable.
y = x^2 - 6x + 5
= x^2 - 6x + 9 - 9 + 5
= (x-3)^2 - 4
The vertex is (3, -4)
so a horizontal line of y = -4 will run through this vertex , that's your "once"
y = -3 will cut it twice
y = -6 will not touch the parabola
in general
y = -4 , once
y = c , c > -4, twice
y = c c < -4 , not at all
a. To determine the coordinates of the turning point of the parabola, we can rewrite the equation in vertex form:
y = a(x - h)^2 + k
Comparing this with the given equation, y = x^2 - 6x + 5, we can see that the coefficient of x^2 is 1, which means a = 1. To complete the square, we need to find h and k.
To find h, we use the formula h = -b/2a. In this case, b = -6 and a = 1:
h = -(-6)/(2*1) = 6/2 = 3
So the x-coordinate of the turning point is 3.
To find k, substitute the value of h back into the original equation:
y = (3)^2 - 6(3) + 5
y = 9 - 18 + 5
y = -4
Therefore, the coordinates of the turning point of the parabola are (3, -4).
b. Now, let's consider the line y = c and determine for which values of c the line will intersect the parabola.
i. To find the values of c for which the line y = c will intersect the parabola twice, we need to find values of x where the quadratic equation has two distinct roots. This occurs when the discriminant (b^2 - 4ac) is greater than 0.
In our case, since the equation of the parabola is y = x^2 - 6x + 5, we need to set the discriminant to be greater than 0:
(-6)^2 - 4(1)(5) > 0
36 - 20 > 0
16 > 0
Since 16 is indeed greater than 0, the line y = c will intersect the parabola twice for any value of c.
ii. To find the values of c for which the line y = c will intersect the parabola once, we need to find values of x where the quadratic equation has one root. This occurs when the discriminant (b^2 - 4ac) is equal to 0.
In our case, we need to set the discriminant to be equal to 0:
(-6)^2 - 4(1)(5) = 0
36 - 20 = 0
16 = 0
However, 16 is not equal to 0. Therefore, the line y = c will not intersect the parabola exactly once.
iii. To find the values of c for which the line y = c will not intersect the parabola at all, we need to find values of x where the quadratic equation has no real roots. This occurs when the discriminant (b^2 - 4ac) is less than 0.
In our case, we need to set the discriminant to be less than 0:
(-6)^2 - 4(1)(5) < 0
36 - 20 < 0
16 < 0
However, 16 is not less than 0. Therefore, the line y = c will intersect the parabola for any value of c, and not at all.
To determine the coordinates of the turning point of the parabola with equation y = x^2 - 6x + 5, we can use the vertex formula.
a. The vertex formula for a parabola with equation y = ax^2 + bx + c is given by:
x = -b / (2a)
y = f(x) = a(x-h)^2 + k
In this case, a = 1, b = -6, and c = 5. Plugging these values into the vertex formula, we get:
x = -(-6) / (2*1)
x = 6 / 2
x = 3
To find the y-coordinate, we substitute x = 3 into the equation:
y = (3)^2 - 6(3) + 5
y = 9 - 18 + 5
y = -4
Therefore, the turning point of the parabola is at coordinates (3, -4).
b. Now, let's analyze the line y = c intersecting the parabola for different values of c.
i. For the line to intersect the parabola twice, the discriminant of the quadratic equation formed by setting the line equal to the parabola must be greater than 0. This indicates that the line intersects the parabola at two distinct points.
For the equation x^2 - 6x + 5 = c, the discriminant is:
D = b^2 - 4ac
D = (-6)^2 - 4(1)(c - 5)
D = 36 - 4(c - 5)
D = 36 - 4c + 20
D = -4c + 56
For two distinct intersections, D > 0. Therefore, -4c + 56 > 0.
Solving this inequality, we get:
-4c + 56 > 0
-4c > -56
c < 14
So, for the line y = c to intersect the parabola twice, c must be less than 14.
ii. For the line to intersect the parabola once, the discriminant must be equal to 0. This indicates that the line touches the parabola at one point.
Setting the discriminant equal to 0, we have:
-4c + 56 = 0
-4c = -56
c = 14
Therefore, for the line to intersect the parabola once, c must equal 14.
iii. For the line to not intersect the parabola at all, the discriminant must be less than 0. This means that the line is either completely above or below the parabola.
Setting the discriminant less than 0, we have:
-4c + 56 < 0
-4c < -56
c > 14
So, for the line to not intersect the parabola at all, c must be greater than 14.
In summary:
i. The line y = c intersects the parabola twice if c < 14.
ii. The line y = c intersects the parabola once if c = 14.
iii. The line y = c does not intersect the parabola at all if c > 14.