Quadratic Functions in Vertex form

Key Concepts

• Explain the graph of 𝒈(𝒙)=x2+k
• Explain the graph of 𝒈(𝒙)=(x-h)2
• Define the vector form of a quadratic function
• Graph using vertex form of a quadratic function

Vertex form of the quadratic function 

The function f(x) = a(x−h)²+k where a≠0 is called the vertex form of the quadratic function. 

• The vertex of the graph g is (h, k). 
• The graph of f(x) = (x−h)²+k is a translation of the function f(x) = x² that is translated in h units horizontally and kk units vertically. 
• The value of a does not affect the location of the vertex. 

Graph of g(x) = x² + k

• The value of k in g(x) = x²+k translates the graph of parent function f, vertically k units. 
• The value of k does not affect the axis of symmetry. 

Graph of g(x) = (x−h)²

• The value of h in g(x) = (x−h)² translates the graph of parent function f, horizontally h units. 

• The vertex of the graph g is (0, h). 

• The value of h translates the axis of symmetry. 

Graph using vertex form of the quadratic function 

• Step 1: Plot the vertex and axis of symmetry. 
• Step 2: Evaluate the function to find other points. 
• Step 3: Draw the parabola through the points. 

Exercise

1. Identify the vertex and axis of symmetry of graph g(x) = x²+2.

2. Write the function in vertex form.

3. Sketch the graph of f(x) = 0.5(x + 2)² + 2.

4. Write the function in vertex form.

5. Identify the vertex and axis of symmetry of graph g(x) = -0.75(X – 5)² + 6.

Concept Map

• The function f(x) = a(x−h)²+k where a≠0 is called the vertex form of the quadratic function. 
• The vertex of the graph g is (h, k)
• The graph of f(x) = (x−h)²+k is a translation of the function f(x) = x² that is translated h units horizontally and k units vertically. 

What have we learned

• The function f(x) = a(x−h)²+k where a≠0 is called the vertex form of the quadratic function.