Graph f(x)=2(x+1)^2+4 Graph f(x)={2 if x>1 -5-x if x<1
What is your question on these three graphs?
How do I graph them?
take the first: y=2(x+1)^2+4
make a table of values
x,y pick some x values,find y
0,6
-1,4
1,12
2,22
-2,6
plot the points, connect them.
one the last one, you have two different regions,above 1, and below 1 plot each on its own side.
To graph the function f(x) = 2(x+1)^2 + 4, follow these steps:
1. Start by identifying the key characteristics of the function:
- The function is a quadratic function, which means it has a parabolic shape.
- The coefficient of x^2 is positive, indicating an upward parabola.
- The function is shifted 1 unit to the left and 4 units upward.
2. Plot the vertex:
- The vertex of the parabola can be found by using the formula x = -b/2a, where a and b are coefficients of the quadratic term and linear term, respectively.
- In this case, a = 2 and b = 2. Plugging these values into the formula, we get x = -2/(2*2) = -1/2.
- To find the y-coordinate of the vertex, substitute x = -1/2 back into the equation: f(-1/2) = 2(-1/2+1)^2 + 4 = 2(1/2)^2 + 4 = 2(1/4) + 4 = 1/2 + 4 = 4.5.
- So, the vertex of the parabola is (-1/2, 4.5).
3. Determine the x-intercepts (if any):
- To find the x-intercepts, set f(x) = 0 and solve for x.
- 2(x+1)^2 + 4 = 0
- (x+1)^2 = -2
- There are no real solutions to this equation since a square cannot be negative.
- Therefore, there are no x-intercepts in this case.
4. Determine the y-intercept:
- To find the y-intercept, set x = 0 and evaluate f(0).
- f(0) = 2(0+1)^2 + 4 = 2(1)^2 + 4 = 2 + 4 = 6.
- So, the y-intercept is (0, 6).
5. Plot additional points:
- Choose some x-values, both positive and negative, and calculate the corresponding y-values to plot additional points.
- For example, when x = -2: f(-2) = 2(-2+1)^2 + 4 = 2(1)^2 + 4 = 2 + 4 = 6. This gives us the point (-2, 6).
- You can choose more x-values following the same process to get more accurate points.
6. Draw the graph:
- Using the plotted points, draw a smooth curve that passes through the vertex, the y-intercept, and any additional points.
To graph the piecewise function f(x) = {2 if x > 1, -5 - x if x < 1}, follow these steps:
1. Identify the key characteristics of the function:
- The function has two different branches depending on the value of x.
- For x > 1, the function is equal to 2.
- For x < 1, the function is equal to -5 - x.
2. Plot the first branch (x > 1):
- Choose a value of x greater than 1, such as x = 2.
- The corresponding y-value is 2.
- Plot the point (2, 2).
3. Plot the second branch (x < 1):
- Choose a value of x less than 1, such as x = 0.
- The corresponding y-value is -5 - x = -5 - 0 = -5.
- Plot the point (0, -5).
4. Draw the graph:
- For x > 1, draw a horizontal line at y = 2 passing through the point (2, 2).
- For x < 1, draw a straight line with a slope of -1 passing through the point (0, -5).
The resulting graph will have a horizontal line at y = 2 for x greater than 1 and a line with a slope of -1 for x less than 1.