Week 8 – Quadratic Functions and Graphs

Posted on by shannonf2015

I have learned many things about graphing such as what information we can take out to determine how a graph looks like of a quadratic function. I have also learned the most important information to determine the equation of a quadratic function and how to draw it as a graph.

Parabola: graph of every quadratic function which is a curve

General Form: y=ax^2+bx+c

Standard Form: y=a{(x-p)}^2+q

Vertex : highest or lowest point; (p,q)

Minimum Point: when the graph opens up (The coefficient next to x^2 is positive)

Maximum Point: when the graph opens down (The coefficient next to x^2 is negative)

Axis of Symmetry: intersects the parabola at the vertex

Domain : All possible values for x (always the element of real numbers)

Range: All possible values for y

X intercepts: zeros of function (values of x when the function is 0, when y=0)

Pattern of parent function, y=x^2: 1,3,5,7,9…

Throughout the week we have different types of transformations:

Parent Function : y=x^2

y=x^2+q : depending on the value of q, the image of the graph y=x^2 moves a vertical translation (moves however many times up or down) while keeping the same points, size and just sliding the vertex. This quadratic equation is therefore congruent to the parent function, y=x^2, because it does not change, it just moves up or down.

y={(x-p)}^2 : depending on the value of p, the image of the graph y=x^2 moves a horizontal translation (however many times to the right or left).

  • {(x-3)}^2 —– when the sign is negative, the vertex of the graph moves to the right
  • {(x+3)}^2 —– when the sign is positive, the vertex of the graph moves to the left

This is also congruent to the parent function.

y={ax}^2: the graph will stretch vertically when a > 1.

When the value of a is between 0 and 1 (a fraction), the graph will compress vertically : 0 < a < 1

This transformation is not congruent to the parent function.

When these transformations are all combined, the equation becomes standard form. 

Let’s look at some examples of using the standard form to determine an equation of a graph.

 

1) Vertex: look at where the highest or lowest point is and write down the coordinates.

(4,1)

2) Plug in the p and q values from the vertex

y=a{(x-p)}^2+q

 

y=a{(x-4)}^2+1

3) Find a point on the graph and use the coordinates to replace x and y.

I’ll use the point : (3, 4)

4=a{(3-4)}^2+1

4) Isolate a

  • 4=a{(3-4)}^2+1
  • 4=a(-1)^2+1
  • 3=a(-1)^2
  • 3 = a

The equation is : y=3(x-4)^2+1

 

Let’s look at how we can determine the domain and range and the intercepts by the quadratic function:

Example: y=3{(x-2)}^2+1

It is always important to first find the vertex as it is the essential piece of information to be able to determine the form of the graph.

y=3{(x-p)}^2+q
  1. Vertex: remember the vertex is (p,q). In our example, p is -2 which becomes +2 because when you place -2 in the standard form, it will become positive with both negatives:
  • y=3{(x-(-2))}^2+q
  • y=3{(x+2)}^2+q

q is the y coordinate of the vertex (not the y-intercept like in general form) which will be 1.

Our vertex is (2, 1)

2. Domain: x∈R – Domain is always the element of real numbers. 

3. Range: we should think about the direction of opening to be able to tell the range. Since the coefficient is +3, the graph opens up. This also means that the vertex is a minimum point with y-coordinate 1.

Therefore, y ≥ 1

4. Direction of opening: Opens up

5. Equation of the axis of symmetry: This is the line that cuts through the vertex.

AOS: x = 2

6. The intercepts:

To find the y-intercept, x=0. Replace x with zero in the equation and solve.

y=3{(0-2)}^2+1

y = 13

To find the x-intercept, y=0. Replace y with zero in the equation and solve.

  • 0=3{(x-2)}^2+1
  • -1=3{(x-2)}^2
  • -\frac{1}{3}={(x-2)}^2

This equation has no solution, so there are no x-intercepts.