Quadratic Functions and Vertex Form
- Quadratic equations can be expressed in different forms:
- Standard Form: Key point is the y-intercept.
- Factored Form: Key points are the x-intercepts (zeros).
- Conversion between Standard Form and Factored Form has been previously covered.
- Introduction to Vertex Form.
- Exploration of Vertex Form using Marbleslides on Desmos.
- Ex. 1 Convert y = (x - 5)^2 to Standard Form.
- Vertex: (5, 0)
- y-int.: (0, 25)
- Ex. 2 Convert y = (x + 3)^2 - 4 to Standard Form.
- Vertex: (-3, -4)
- y-int.: (0, 5)
- Ex. 3 Convert y = -2(x - 4)^2 + 1 to Standard Form.
- Vertex: (4, 1)
- y-int.: (0, -31)
- Ex. 4 Convert y = x^2 - 14x + 3 to Vertex Form.
- Vertex: (7, -46)
- y-int.: (0, 3)
- Ex. 5 Convert y = x^2 - 12x to Vertex Form.
- Vertex: (6, -36)
- y-int.: (0, 0)
- Ex. 6 Convert y = x^2 + 12x + 11 to Vertex Form.
- Vertex: (-6, -25)
- y-int.: (0, 11)
Practice
- Answers provided for various quadratic relations including vertex, y-intercept, and standard form.
- vertex: (4,0) y-int: (0, 16), standard form: N/A
- c) vertex: (-3,-2), y-int: (0, 7), standard form: N/A
- d) vertex: (-1,5) y-int: (0, 5.5), standard form: N/A
- f) vertex: (1, 7), y-int: (0, 6), standard form: N/A
- g) vertex: (4, -5), y-int: (0, 27), standard form: N/A
- h) vertex: (-4,-2), y-int: (0, -50), standard form: N/A
- Transformations of graphs are changes to the basic graph.
- A translation is a movement up, down, right, or left.
- y = x^2
- y = x^2 - 3
- y = x^2 + 4
- On the graph of y = x^2 + k, the k represents a vertical translation of k units up or down. It impacts the y-values.
- y = (x - 3)^2
- y = (x + 4)^2
- On the graph of y = (x - h)^2, the h represents a horizontal translation of h units to the right or left. It impacts the x-values.
- Vertex form: y = (x - h)^2 + k
- On the graph of y = (x - h)^2 + k, the vertex is the point (h, k)
- A parabola can be vertically stretched or compressed by changing the “a” value in front of the brackets in vertex form: y = a(x - h)^2 + k
- Stretch factor:
- If the a value is larger than 1, it is called a vertical stretch.
- If the a value is between 0 and 1, it is called a vertical compression.
- If the a value is negative, the transformation is called a reflection in the x-axis.
- The graph of y = ax^2 is vertically stretched or compressed by the a value, meaning that each y-value from y = x^2 is multiplied by a.
Determining Stretch Value Given a Graph
Practice
- Write an equation for the parabola with vertex at (-3, 0), opening downward, and with a vertical stretch of factor 2: y = -2(x + 3)^2
- Write an equation for the parabola with vertex at (4, -1), opening upward, and with a vertical compression of factor 0.3: y = 0.3(x - 4)^2 - 1
- The graph of y = x^2 is stretched vertically by a factor of 3 and then translated 2 units to the left and 1 unit down: y = 3(x + 2)^2 - 1
- The graph of y = x^2 is reflected in the x-axis, compressed vertically by a factor of \frac{1}{2}, and then translated 2 units upward: y = -\frac{1}{2}x^2 + 2
- Find an equation for the parabola with vertex (1, 4) that passes through the point (3, 8): y = (x - 1)^2 + 4
- Find an equation for the parabola with vertex (-2, 5) and y-intercept 1: y = -(x + 2)^2 + 5
- Key Concepts for y = a(x - h)^2 + k
- The vertex of the parabola is (h, k), representing a horizontal translation of h units and a vertical translation of k units relative to the graph of y = x^2.
- The axis of symmetry of the parabola is the vertical line through the vertex with equation x = h.
- a indicates the vertical stretch or compression factor relative to the graph of y = x^2.
- If a > 0, the parabola opens upward, and the vertex is the minimum point on the graph.
- If a < 0, the parabola opens downward, and the vertex is the maximum point on the graph.
- Property, Vertex, Axis of symmetry, compression factor, Direction of opening
Practice Questions
- Write an equation for the parabola with vertex at (2, 3), opening upward, and with no vertical stretch: y = (x - 2)^2 + 3
- Write an equation for the parabola with vertex at (-3, 0), opening downward, and with a vertical stretch of factor 2: y = -2(x + 3)^2
- Write an equation for the parabola with vertex at (4, -1), opening upward, and with a vertical compression of factor 0.3: y = 0.3(x - 4)^2 - 1
- The graph of y = x^2 is stretched vertically by a factor of 3 and then translated 2 units to the left and 1 unit down: y = 3(x + 2)^2 - 1
- Standard form of a quadratic relation: y = ax^2 + bx + c.
- In standard form, the only key point that we can easily see is the y-intercept.
- To convert from standard form back to vertex form we need to remake the (x - h)^2 part.
- We call this algebraic process completing the square.
- We need to make sure we don’t change the original equation, so we need to subtract the value we added to make the perfect square.
Practice Questions
- Determine the value of c that makes each expression a perfect square.
- Rewrite each relation in the form y = a(x - h)^2 + k by completing the square.
- Find the vertex of each quadratic relation by completing the square.
- Find the vertex of each parabola. Sketch the graph, labeling the vertex, the axis of symmetry, and two other points.
- Find the two missing values (b, c, and/or h) in each equation.
- Example 1: Convert from VERTEX to STANDARD and from STANDARD to VERTEX:
- y = 2x^2 + 12x + 11 \Rightarrow y = 2(x + 3)^2 - 7
- Example 2: Convert from VERTEX to STANDARD and from STANDARD to VERTEX:
- y = -3x^2 + 12x - 13 \Rightarrow y = -3(x - 2)^2 - 1
- Example 3:
- The path of a ball is modeled by the equation y = x^2 + 2x + 3, where x is the horizontal distance, in metres, from a fence and y is the height, in metres, above the ground.
- What is the maximum height of the ball, and at what horizontal distance does it occur?
- Sketch a graph to represent the path of the ball.
- The y-intercept is a point where the x-value is 0. The coordinates look like: (0, y)
- The x-intercepts are points where the y-value is 0. The coordinates look like: (x, 0)
- The x-intercepts are also called the zeros of the parabola.
- Example 1: y = \frac{1}{2}(x - 5)^2 - 18
- y-intercept:
- x-intercept(s):
Review
- Rewrite each relation in the form y = a(x - h)^2 + k by completing the square.
- Find the vertex of each parabola. Sketch the graph, labeling the vertex, the axis of symmetry, and two other points.
- Determine an equation to represent each parabola.
- Sketch a graph of each parabola. Label the coordinates of the vertex and the equation of the axis of symmetry.
- Sketch the graph of each parabola. Describe the transformation from the graph of y = x^2.
- Copy and complete the table for each parabola. Replace the heading for the second column with the equation for the parabola. Then, sketch each parabola.
Quadratic Word Problems
- The Empire State Building is 1250 feet tall. An object is thrown upward from the building in the shape h = -16(t - 1.5)^2 + 1280, where h is in feet and t in seconds.
- a) What is the maximum height the object reaches?
- b) How long does it take to reach this height?
- c) When will the object hit the ground?
- d) Was the ball thrown from the roof?
- e) State the domain and range.
- A water balloon is catapulted into the air so that its height h, in metres, after t seconds is h = -5(t - 2.75)^2 + 40.
- a) How high is the balloon after 1 second?
- b) For how long is the balloon more than 30 m high?
- c) What is the maximum height of the balloon?
- d) When will the balloon burst as it hits the ground?
- e) How high off of the ground was the balloon launched?