section 4.1 – quadratic functions and translations

Section 4.1 – Quadratic Functions and Translations


In the Solve It, you used a parabolic shape of the horse’s jump. A parabola is the graph of a

quadratic function, which you can write in the form y = ax2 + bx + c, where a cannot be 0.

Essential Understanding

The graph of ANY quadratic function is a transformation of the graph of the parent

quadratic function y = x2.


The vertex form of a quadratic function is

f(x) = a(x – h)2 + k, where a cannot be 0.

The axis of symmetry is a line that divides the parabola into two mirror images. The equation of the axis of symmetry is x = h. The vertex of the parabola is (h, k) the intersection of the parabola

and its axis of symmetry.


Problem 1:

What is the graph of f(x) = ½x2


Problem 1:

What is the graph of f(x) = -1/3x2


Graphs of y = ax2 and y = -ax2 are reflections of each other across the x-axis.

Increasing |a| stretches the graph vertically and narrows it horizontally. Decreasing |a|

compresses the graph vertically and widens it horizontally.


Problem 2:

Graph each function. How is each graph a translation of f(x) = x2

g(x) = x2 – 5


Problem 2:


g(x) = (x – 4)2


Problem 2:


g(x) = x2 + 3


Problem 2:


g(x) = (x + 1)2


The vertex form, f(x) = a(x – h)2 + k,

gives you information about the graph of f without drawing the graph.

If a > 0, k is the minimum value of the function.

If a < 0, k is the maximum value.


Problem 3:

For y = 3(x – 4)2 – 2, what are the vertex, the axis of symmetry, the maximum or minimum value,

the domain and range?


Problem 3:

For y = -2(x + 1)2 + 4, what are the vertex, the axis of symmetry, the maximum or minimum

value, the domain and range?


You can use the vertex of a quadratic function,

f(x) = a(x – h)2 + k, to transform the graph of the parent function f(x) = x2.

• Stretch or compress the graph of f(x) = x2 vertically by a factor of |a|

•If a < 0, reflect the graph across the x-axis

•Shift the graph |h| units horizontally and |k| units vertically


Problem 4:

What is the graph of f(x) = -2(x – 1)2 + 3?


Problem 4:

What is the graph of f(x) = 2(x +2)2 – 5 ?


Problem 5:

Write an equation of the parabola through

Vertex(1, 2)

Point(2, -5)


Problem 5:


Vertex(-3, 6)

Point(1, -2)


Problem 5:


Vertex(0, 5)

Point(1, -2)


Problem 6:

A gardener is putting a wire fence along the edge of his garden to keep animals from eating his

plants. If he has 20 meters of fence, what is the largest rectangular area he can enclose?


Problem 6:

Write a quadratic function to represent the areas of all rectangles with a perimeter of 36 feet.

Graph the function and describe the rectangle that has the largest area.

section 4.1 – quadratic functions and translations

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