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.

Section 4.1 – Quadratic Functions and Translations 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.

Section 4.1 – Quadratic Functions and Translations

Section 4.1 – Quadratic Functions and Translations Problem 1: What is the graph of f(x) = ½x2

Section 4.1 – Quadratic Functions and Translations Problem 1: What is the graph of f(x) = -1/3x2

Section 4.1 – Quadratic Functions and Translations 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.

Section 4.1 – Quadratic Functions and Translations

Section 4.1 – Quadratic Functions and Translations Problem 2: Graph each function. How is each graph a translation of f(x) = x2 g(x) = x2 – 5

Section 4.1 – Quadratic Functions and Translations Problem 2: Graph each function. How is each graph a translation of f(x) = x2 g(x) = (x – 4)2

Section 4.1 – Quadratic Functions and Translations Problem 2: Graph each function. How is each graph a translation of f(x) = x2 g(x) = x2 + 3

Section 4.1 – Quadratic Functions and Translations Problem 2: Graph each function. How is each graph a translation of f(x) = x2 g(x) = (x + 1)2

Section 4.1 – Quadratic Functions and Translations 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.

Section 4.1 – Quadratic Functions and Translations 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?

Section 4.1 – Quadratic Functions and Translations 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?

Section 4.1 – Quadratic Functions and Translations 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

Section 4.1 – Quadratic Functions and Translations

Section 4.1 – Quadratic Functions and Translations Problem 4: What is the graph of f(x) = -2(x – 1)2 + 3?

Section 4.1 – Quadratic Functions and Translations Problem 4: What is the graph of f(x) = 2(x +2)2 – 5 ?

Section 4.1 – Quadratic Functions and Translations Problem 5: Write an equation of the parabola through Vertex(1, 2) Point(2, -5)

Section 4.1 – Quadratic Functions and Translations Problem 5: Write an equation of the parabola through Vertex(-3, 6) Point(1, -2)

Section 4.1 – Quadratic Functions and Translations Problem 5: Write an equation of the parabola through Vertex(0, 5) Point(1, -2)

Section 4.1 – Quadratic Functions and Translations 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?

Section 4.1 – Quadratic Functions and Translations 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.