Sketch the graph of the function: y=(x-2)(x-3)(x+5)

Question

Sketch the graph of the function: y=(x-2)(x-3)(x+5)

1. Determine the end behavior of the function.

2. Find all x-intercepts.

3. Find all y-intercepts.

4. Use the x-intercepts to divide the x axis into intervals.

5. Choose at least one point within each interval to show whether function is positive or negative.

6. Sketch the graph.

 

2. If f(x)=4x-1 and g(x)=5x, what is (fog)(7)?

 

3. What two end behaviors are true of the following function?

 h(t)=2t^4 - t^3 +6t^2 +2t-8

 

To solve the following quadratic by completing the square, what value would you add to both sides of the equation?

x^2 - 6x =7

Answer 1

Problem 1:

 

1) Write out the function in standard form, i.e. ax³ + bx² + cx + d = 0

 

Multiply (x-2)*(x-3) first, the multiply what you get by (x+5)

 

As x goes to infinity, your function does, too, because the coefficient of the cubic term is positive

 

As x goes to negative infinity, the function goes to negative infinity

 

2) The x intercepts are when y = 0. Conveniently, you can find those from the original equation.

 

They are 2, 3, and -5

 

3) y intercept is when x=0. This is easier to do with the standard equation equation you found in part (1)

 

4) This is clear from the instructions. The segments are (-∞, -5) (-5, 2) (2,3) (3, ∞)

 

5) Pick a point and evaluate

 

6) Use the zeroes and points you just found to make a rough sketch.

 

Problem 2:

 

You are asked to find the function f(g(x)). So plug g(x) into f(x) to find the answer

 

fog(x) = 4 * (5x) - 1

 

Evaluate at x =7

 

Problem 3:

 

This is a quartic function with a positive coefficient. It goes to infinity on both ends (for the same reason that the quadratic does - the even power cancels out the negative)

 

Problem 5:

 

Complete the square of

 

x² - 6x = 7

 

Use the rule that the coefficient of the x term is divided by 2, then squared.

 

(-6/2)^2 = 9

 

x² - 6x + 9 = 7 + 9

 

(x-3)² = 16

 

I hope this helps!