Graphs of Quadratics

Graphs of Quadratics

Learning Targets

  • I can graph quadratics.

  • I can identify the key characteristics of quadratic graphs.

  • I can relate the graphs to real life characteristics.

Warm Up

  1. Solve for x in both equations below:

    • x - 5 = 0

    • 3x - 1 = 0

  2. Multiply the following binomials together:

    • (x + 1)(x - 2)

Quadratic Functions

A quadratic function is a type of polynomial function with a highest degree of 2. A graph of a quadratic function makes the shape of a parabola.

Forms of Quadratic Functions

  1. Standard Form Quadratic:
    y = ax^2 + bx + c
    Where c is the y-intercept.

  2. Factored Form Quadratic:
    y = (x - R1)(x - R2)
    Where x = R1 and x = R2 are the zeros of the function (the x-intercepts).

  3. Vertex Form:
    y = a(x - h)^2 + k
    Where (h, k) is the vertex of the parabola.

Leading Coefficients

  • If the ends of the graph are going up, the leading coefficient (LC) is positive.

  • If the ends of the graph are going down, the leading coefficient (LC) is negative.

Parts of a Parabola

  • Vertex Form: The vertex representation of the graph.

  • Factored Form: The representation of the zeros of the quadratic.

  • Standard Form: The traditional polynomial expression.

Characteristics of Graphs

Key Characteristics Description

  1. Domain:

    • All of the x values your graph touches. Read from left to right.

  2. Range:

    • All of the y values your graph touches. Read from bottom to top.

  3. Increasing:

    • As you move from left to right, where the graph is going upward.

  4. Decreasing:

    • As you move from left to right, where the graph is going downward.

  5. Positive:

    • As you move from left to right where the graph is above the x-axis (where y values are positive).

  6. Negative:

    • As you move from left to right where the graph is below the x-axis (where y values are negative).

  7. Absolute Maximum:

    • Located at the top of the graph when the ends are going down. This is the highest point the graph will reach.

  8. Absolute Minimum:

    • Located at the bottom of the graph when the ends are going up. This is the lowest point the graph will reach.

  9. End Behavior:

    • Where the function is going as x goes towards negative infinity and positive infinity.

Average Rate of Change

  • This measures how much y is changing over a given interval (x1, x2).

Graph Characteristics Review

Examples: Label the parts of the graph and identify each characteristic.

  • Domain:

  • Range:

  • End Behavior:

    • x o - ext{∞}, y o ext{???}

    • x o ext{∞}, y o ext{???}

  • Positive:

  • Negative:

  • Increasing:

  • Decreasing:

  • Max or Min:

  • Average rate of change over [0, 2]:

You Try:

Label the characteristics based on your graph.

  • Domain:

  • Range:

  • End Behavior:

    • x o - ext{∞}, y o ext{???}

    • x o ext{∞}, y o ext{???}

  • Positive:

  • Negative:

  • Increasing:

  • Decreasing:

  • Max or Min:

  • Average rate of change over [-4, -2]:

Sketching Graphs in Vertex Form

Steps

  1. Identify the vertex and plot it.

  2. Identify the ‘a’ value and use it to plot two more points.

    • + a:

    • - a:

    • Always over 1 on each side.

Examples

  1. Example 1:

  2. Example 2: f(x) = -2(x - 3)^2 - 1

    • Vertex:

    • a:

  3. Example 3: Write the equations of each graph below in vertex form.

    • Equation 1:

    • Equation 2:

Applications of Quadratic Graphs

When we analyze graphs of quadratic functions to represent real life applications, we look into the following areas:

  • Projectile motion (e.g., soccer balls being kicked).

  • Height over time (e.g., rockets launched).

Example 1

Imagine a soccer ball being kicked in a game.

Example 2

The graph below demonstrates the height y in meters of a rocket launched over time x in seconds.

  1. At what height does the rocket initially start?

  2. What does this mean?

  3. How long is the rocket in the air?

  4. What is the greatest height that the rocket reaches?

  5. What is the domain of the rocket?

  6. What is the range of the rocket?

  7. Is the rocket moving faster from 0 seconds to 1 second, or from 7 to 8 seconds? Show work to support your answer.