Graphing Quadratic Functions: Unveiling Standard Form with Intercepts

Algebra 2 Grades High School 15:38 Video
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Lesson Description

Master graphing quadratic functions in standard form by finding the vertex, intercepts, and using a table of values. This lesson builds on previous knowledge and introduces a complete method for accurate graphing.

Video Resource

Graphing Quadratic Functions Algebra 2 | Standard Form with Intercepts

Kevinmathscience

Duration: 15:38
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Key Concepts

  • Standard form of a quadratic equation (ax² + bx + c)
  • Vertex of a parabola and its calculation (-b/2a)
  • X-intercepts (roots) and Y-intercepts of a quadratic function
  • Using factoring to find the x-intercepts

Learning Objectives

  • Students will be able to identify the vertex of a quadratic function given in standard form.
  • Students will be able to calculate and identify the x and y intercepts of quadratic functions.
  • Students will be able to graph quadratic functions accurately using the vertex, intercepts, and a table of values (optional).

Educator Instructions

  • Introduction (5 mins)
    Briefly review the standard form of a quadratic equation. Remind students about finding the vertex using the formula x = -b/2a and using it in a table of values. Introduce the concept of x and y intercepts.
  • Vertex Calculation (10 mins)
    Demonstrate how to find the vertex of a quadratic function in standard form using the formula x = -b/2a. Emphasize the importance of correctly identifying 'a', 'b', and 'c' values. Work through an example problem. Then explain how to plug the x-value into the original equation to solve for the corresponding y-value.
  • Finding Y-Intercepts (5 mins)
    Explain that the y-intercept is the point where the graph crosses the y-axis (x=0). Show how to find the y-intercept by substituting x = 0 into the quadratic equation. Highlight that the y-intercept is simply the 'c' value in the standard form equation.
  • Finding X-Intercepts (15 mins)
    Explain that the x-intercepts are the points where the graph crosses the x-axis (y=0). Show how to find the x-intercepts by setting the quadratic equation equal to zero and solving for x. Review factoring techniques for quadratic equations. Emphasize checking for common factors before factoring the trinomial. If factoring is not possible, briefly mention (but don't demonstrate) the quadratic formula.
  • Graphing with Vertex and Intercepts (10 mins)
    Guide students on how to plot the vertex and intercepts on a coordinate plane. Discuss how the vertex and intercepts provide a good initial sketch of the parabola. Remind students about the symmetry of parabolas and how the vertex is the axis of symmetry. Optional: Briefly show how to create a table of values to get additional points for a more accurate graph if needed.
  • Practice Problems (10 mins)
    Provide students with practice problems to work on individually or in pairs. Circulate to answer questions and provide assistance.

Interactive Exercises

  • Online Graphing Tool Activity
    Use an online graphing calculator (e.g., Desmos) to graph quadratic functions given in standard form. Students can manipulate the values of 'a', 'b', and 'c' and observe how the graph changes in real-time. Have students compare their work with the calculator to ensure accurate graphing.
  • Intercept Scavenger Hunt
    Provide students with a set of quadratic equations. Students must find the x and y intercepts for each equation and then 'hunt' for those points on a pre-made coordinate grid. The grid should have the points marked but unlabeled; students must match the intercepts to the correct points on the grid.

Discussion Questions

  • How does the sign of 'a' in the standard form of a quadratic equation affect the graph?
  • What are the advantages and disadvantages of using a table of values versus relying solely on the vertex and intercepts for graphing?
  • How do the x-intercepts relate to the solutions of the quadratic equation?

Skills Developed

  • Algebraic manipulation
  • Problem-solving
  • Analytical thinking
  • Graphing skills

Multiple Choice Questions

Question 1:

What is the standard form of a quadratic equation?

Correct Answer: y = ax² + bx + c

Question 2:

How do you find the x-coordinate of the vertex of a quadratic equation in standard form?

Correct Answer: x = -b/2a

Question 3:

The y-intercept of a quadratic equation in standard form is equal to which value?

Correct Answer: c

Question 4:

To find the x-intercepts of a quadratic equation, you set which variable equal to zero?

Correct Answer: y

Question 5:

What method is used to determine what values of x, where y = 0?

Correct Answer: Factoring

Question 6:

What are the x-intercepts also known as?

Correct Answer: The roots

Question 7:

The vertex is found at (-2, 4). What is the axis of symmetry?

Correct Answer: x = -2

Question 8:

If a = -2, what direction does the quadratic function open?

Correct Answer: Downward

Question 9:

If you can't factor a quadratic equation to find the x-intercepts, what other method can you use?

Correct Answer: All of the above

Question 10:

The axis of symmetry always passes through which point on the parabola?

Correct Answer: The vertex

Fill in the Blank Questions

Question 1:

The formula to find the x-coordinate of the vertex is x = _______.

Correct Answer: -b/2a

Question 2:

The y-intercept occurs when the x-value is _______.

Correct Answer: 0

Question 3:

The x-intercepts occur when the y-value is _______.

Correct Answer: 0

Question 4:

The line of symmetry of a quadratic function always passes through the _______.

Correct Answer: vertex

Question 5:

In the standard form equation y=ax²+bx+c, the y-intercept is represented by the variable _____.

Correct Answer: c

Question 6:

The x-intercepts are also referred to as the _______ of the quadratic equation.

Correct Answer: roots

Question 7:

If a quadratic has x-intercepts at x = 2 and x = -3, then (x - 2) and (x + 3) are the _______ of the quadratic.

Correct Answer: factors

Question 8:

Finding the _______ can help to graph the quadratic function.

Correct Answer: vertex

Question 9:

Setting y = 0 and solving for x using _______ helps to find x-intercepts.

Correct Answer: factoring

Question 10:

In y = ax² + bx + c, the _______ is positive or negative determines the direction the parabola opens.

Correct Answer: a

Educational Standards

A2.F.1.3:Graph a quadratic function. Identify the domain, range, x- and y-intercepts, maximum or minimum value, axis of symmetry, and vertex using various methods and tools that may include a graphing calculator or appropriate technology. A2.A.1.1:Use mathematical models to represent quadratic relationships and solve using factoring, completing the square, the quadratic formula, and various methods (including graphing calculator or other appropriate technology). Find non-real roots when they exist. A2.A.2.1:Factor polynomial expressions including, but not limited to, trinomials, differences of squares, sum and difference of cubes, and factoring by grouping, using a variety of tools and strategies.

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