5.2: Vertex of a parabola

1. Let \(f(x)=x^2\) and \(g(x)=3(x-1)^2+2\).

1. What is the vertex of \(f(x)\) ?
2. Since \(g(x)\) is a translation of \(f(x)\), use your knowledge of horizontal and vertical shifts to identify the vertex of \(g(x)\).
3. What is the formula for the vertex form of a quadratic function?
4. According to the formula, was the vertex you identified in part (b) correct? Why or why not?

 

 

 

2. Let \(g(x)=(x+3)^2-4\).

1. What is the vertex of \(g(x)\) ?
2. Graph \(g(x)\) using Desmos. What is the axis of symmetry of \(g(x)\) ?
3. How can you find the axis of symmetry for any given quadratic function?
4. Write the function \(g(x)\) in standard form.
5. Explain how the vertical intercept can be found from the standard form.
6. Find the zeros of \(g(x)\).
7. Explain how the axis of symmetry of the parabola is related to the zeros.

 

 

 

3. Let \(f(x)=3 x^2-6 x+3\).

1. What is the vertical intercept of \(f(x)\) ?
2. Find the zeros of \(f(x)\).
3. Write the function \(f(x)\) in vertex form.
4. What is the vertex of \(f(x)\) ?
5. What is the axis of symmetry of \(f(x)\) ?

 

 

 

4. Find a formula for the quadratic function based on the information given.

1.  

   

2. Vertex at \((4,2)\) and \(y\)-intercept of \(y=6\).
3. Vertex at \((4,2)\) and zeros at \(x=-3\) and \(x=11\).

 

 

 

5. Let \(m(n)=\frac{1}{2} n^2\). Write a formula in standard form for each of the transformations listed below. Graph the new function.

1. \(g(n)=m(n)+1\)
2. \(h(n)=m(n+1)\)
    

 

 

6. Let \(f(x)=2 x^2+4 x+4\) and \(g(x)=-2 x^2+4 x+4\).

1. Does \(f(x)\) have an absolute maximum or an absolute minimum? How do you know? What is its coordinate?
2. Does \(g(x)\) have an absolute maximum or an absolute minimum? How do you know? What is its coordinate?
    

 

 

7. If you have a string of length 50 cm ,what are the dimensions of the rectangle of maximum area that you can enclose with your string? Hint: let one side of your string rectangle have a length of \(x \mathrm{~cm}\). Then write an expression for the length of the other side, and use the area formula for a rectangle.

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