2)If the scale factor of a map is 1/8” is 12 miles. How far apart are two towns if they are 3 inches apart on the map?
help with graphing?
1)Graph the function y = 2x2 – 3. You can describe your graph by specifying the vertex, maximum or minimum, axis of symmetry, and 4 points on the parabola, two on each side of the axis of symmetry.
2)You have 32 feet of fencing to enclose a garden. The function A=16x-x2, where x=width, gives you the area of the garden in square feet.
a.What is the maximum area?
b.What width gives you the maximum area?
4) Solve x2 -8x +15 =0 by factoring.
1/8 = .125
so
.125/12 = 3/x
x = 36/.125 = 288 miles
1)Graph the function y = 2x2 – 3. You can describe your graph by specifying the vertex, maximum or minimum, axis of symmetry, and 4 points on the parabola, two on each side of the axis of symmetry.
2 x^2 - 3 = y complete the square
x^2 - 1.5 = .5 y
x^2 = .5 y + 1.5
(x-0)(x-0) = .5 (y + 3)
vertex at (0,-3)
axis of symmetry at x = 0
minimum at vertex
You can do points, x = +/-1 and x = +/- 2
2)You have 32 feet of fencing to enclose a garden. The function A=16x-x2, where x=width, gives you the area of the garden in square feet.
a.What is the maximum area?
b.What width gives you the maximum area?
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perimeter = 2x+2y= 32
or y = (16-x)
A = x (16-x) = 16 x - x^2 sure enough
now same deal, complete the square to find vertex
x^2 - 16 x = -A
x^2 - 16 x + 64 = -A + 64
(x-8)^2 = - (A-64)
vertex at A = 64 and x = 8
4) Solve x2 -8x +15 =0 by factoring.
Hey, did you try ???
(x-5)(x-3) = 0
x = 5 or x = 3
2) To find the distance between two towns on the map, we need to use the scale factor given and apply it to the distance on the map.
Given:
Scale factor = 1/8 inch represents 12 miles on the map
Distance on the map = 3 inches
To find the actual distance between the two towns:
1. Set up a proportion using the scale factor:
(1/8 inch) / (12 miles) = (3 inches) / (x miles) (Let x be the actual distance between the towns in miles)
2. Cross multiply and solve for x:
(1/8)(x miles) = (12 miles)(3 inches)
x/8 = 12*3
x/8 = 36
x = 36*8
x = 288
Therefore, the two towns are 288 miles apart.
For help with graphing:
1) The given function is y = 2x^2 - 3. To graph this function, follow these steps:
a) Find the vertex: The vertex of a parabola in the form y = ax^2 + bx + c can be found using the formula x = -b / (2a). In this case, a = 2 and b = 0 (since there's no x term).
x = -0 / (2 * 2) = 0
Substitute the x-value into the equation to find the y-value:
y = 2(0)^2 - 3 = -3
So, the vertex is (0, -3).
b) Determine whether it opens upward (a > 0) or downward (a < 0). In this case, since a = 2 (which is greater than 0), the parabola opens upward.
c) Find the axis of symmetry: The axis of symmetry is the vertical line that passes through the vertex. In this case, the axis of symmetry is x = 0.
d) Choose some x-values to find corresponding y-values and plot the points. Let's use x = -2, -1, 1, and 2:
For x = -2:
y = 2(-2)^2 - 3
= 8 - 3
= 5
So, one point is (-2, 5).
For x = -1:
y = 2(-1)^2 - 3
= 2 - 3
= -1
Another point is (-1, -1).
For x = 1:
y = 2(1)^2 - 3
= 2 - 3
= -1
Another point is (1, -1).
For x = 2:
y = 2(2)^2 - 3
= 8 - 3
= 5
Another point is (2, 5).
e) Plot the vertex and the other points on a graph. Connect the points to form a smooth curve. Label the axis of symmetry and any other relevant information.
2) To find the maximum area of a garden using 32 feet of fencing and the function A = 16x - x^2, where x represents the width, follow these steps:
a) Find the maximum area: To find the maximum or minimum value of a quadratic function in the form A = ax^2 + bx + c, use the formula x = -b / (2a). In this case, a = -1 and b = 16.
x = -16 / (2 * -1) = -16 / -2 = 8
Substitute the x-value into the equation to find the maximum area:
A = 16(8) - (8)^2 = 128 - 64 = 64 square feet
So, the maximum area is 64 square feet.
b) Find the width that gives the maximum area: The width is given by the x-value from step a, which is 8 feet. So, the width that gives the maximum area is 8 feet.
To solve x^2 - 8x + 15 = 0 by factoring, follow these steps:
1. Rewrite the equation in the form ax^2 + bx + c = 0:
x^2 - 8x + 15 = 0
2. Factor the quadratic expression:
(x - 3)(x - 5) = 0
3. Set each factor equal to zero and solve for x:
x - 3 = 0 or x - 5 = 0
x = 3 or x = 5
Therefore, the solutions to the equation x^2 - 8x + 15 = 0 are x = 3 and x = 5.