Quadratic Relationships
Unit 1: Graphing Vertex Form
Learning Goals
- you should be able to tell the difference between a quadratic relation and a linear relation ( first and second differences)
- you should be able to describe the characteristics of a parabola. (i.e Vertex, Axis of symmetry)
- you should be able to describe transformations of a quadratic relation and graph it in vertex form
Summary Of The Unit
- Vertex Equation= a(x-h)^2 + k
- The vertex in the vertex form is (h,k)
- Axis of symmetry is a line that goes through the vertex (x=h)
- Optimal value is the highest point of the parabola (y=k)
- A in the equation tells you the direction of opening and whether the parabola is compressed or stretched. If a>0, the parabola opens up and if a<0 the parabola opens down. If -1< a < 1 the parabola is vertically compressed. If a>1 or a<-1 the parabola is vertically stretched.
- K is the vertical translation which tells you how much the parabola is translated up or down. If k>0, the vertex moves up by k units and if k< 0, the vertex moves down by k units.
- H is the horizontal translation which tells you how much the parabola translated left or right. If h>0, the vertex moves to the right h units and if h<0 the vertex moves to the left h units.
- Step Pattern
- Mapping Notation
- To find the y-intercept, set x= 0 and solve for y
- To solve for x, set y=0 and solve for x or expand and simplify to get the standard form, then use the quadratic formula
Graph
Here is a graph that has 2 x-intercepts.
How to Solve a word Problem using Vertex Form
Here is our first question! This is really easy and the answer is given in the equation. If we were to graph this equation the maximum height would be the highest point on the parabola. It would be the highest point because the graph it self is opening downward. The answer to this question is simple. It would be your K value which is 182 metres. This would be your maximum height because it is the K is the optimal value (highest point on the parabola). Vertex= (h,k). In any word problem question, do not forget to use the word therefore or end off with a concluding sentence just to get those full marks!
This question is very similar to the first one. The answer to this question would be the "h" value in this equation. The answer would be 6 because it is the h value and h represents the horizontal distance of the maximum or minimum point of the parabola. The reason why the answer is 6 seconds and not -6 seconds is because when the h value comes out of the bracket it has the opposite sign.
As we know h represents the height as said in the question. For this question, we are going to find out what h is.
Our equation is h= -4.9(t-6)^2 + 182. To find h, we are going to replace t with 0. and solve the equation.
I will now solve it.
Step 1: h= -4.9 (0-6)^2 + 182
Step 2: h= -4.9(-6)^2 + 182
Step 3: h= -4.9(36) + 182
Step 4: h= -176.4 + 182
Step 5: h= 5.6 m
Therefore, the height above the ground was 5.6m.
And thats how you solve a word problem using vertex form!