1. Introduction
Imagine the path of a basketball as it leaves a player’s hand and swooshes through the net. Or, imagine the path of an Olympic diver as she arcs away from the diving board, slips deep into the water, and resurfaces. Both traveling paths make a U-shaped curve known as a parabola. A parabola is the U-shaped graph of a quadratic function, which is an equation with a degree of 2. Formally, a quadratic function is a function that can be written in the form f(x) = ax2 + bx + c, where a ≠ 0. Quadratics can be used to model varying situations. In this lesson, you will learn about and practice identifying the important points on a parabola and use those points to sketch the corresponding graph.
5.3.1: Creating and Graphing Equations Using Standard Form

2. Key Concepts
The standard form of a quadratic function is f(x) = ax2 + bx + c, where a is the coefficient of the quadratic term, b is the coefficient of the linear term, and c is the constant term.
The x-intercepts of the graph of a quadratic function are the points at which the graph crosses the x-axis, and are written as (x, 0).
The y-intercept of the graph of a quadratic function is the point at which the graph crosses the y-axis and is written as (0, y).
The graph on the following slide shows the location of the parabola’s y-intercept.
5.3.1: Creating and Graphing Equations Using Standard Form

3. Key Concepts, continued
5.3.1: Creating and Graphing Equations Using Standard Form

4. Key Concepts, continued
The equation of the parabola is y = x2 – 2x – 4.
Note that the y-intercept of this equation is –4 and the constant term of the quadratic equation is –4.
The y-intercept of a quadratic is the c value of the quadratic equation when written in standard form.
The axis of symmetry of a parabola is the line through the vertex of a parabola about which the parabola is symmetric.
5.3.1: Creating and Graphing Equations Using Standard Form

5. Key Concepts, continued
5.3.1: Creating and Graphing Equations Using Standard Form

6. Key Concepts, continued
The axis of symmetry extends through the vertex of the graph.
The vertex of a parabola is the point on a parabola where the graph changes direction, (h, k), where h is the x-coordinate and k is the y-coordinate.
The equation of the axis of symmetry is
In the example above, the equation of the axis of symmetry is x = 1 because the vertical line through 1 is the line that cuts the parabola in half.
5.3.1: Creating and Graphing Equations Using Standard Form

7. Key Concepts, continued
The axis of symmetry is sometimes written as
where h is the x-coordinate of the vertex.
The vertex of a quadratic lies on the axis of symmetry, as shown in the graph on the next slide.
The formula is not only used to find the
equation of the axis of symmetry, but also to find the x-coordinate of the vertex.
5.3.1: Creating and Graphing Equations Using Standard Form

8. Key Concepts, continued
5.3.1: Creating and Graphing Equations Using Standard Form

9. Key Concepts, continued
To find the y-coordinate, substitute the value of x into
the original function,
The vertex gives information about the maximum or minimum value of a quadratic.
The maximum is the largest y-value of a quadratic and the minimum is the smallest y-value.
One might see these terms in real-world situations such as maximizing profit and minimizing cost, among others.
5.3.1: Creating and Graphing Equations Using Standard Form

10. Key Concepts, continued
From the equation of a function in standard form, you can determine if you have a maximum or minimum based on the sign of the coefficient of the quadratic term, a.
If a > 0, the parabola opens up and therefore has a minimum value.
If a < 0, the parabola opens down and therefore has a maximum value.
5.3.1: Creating and Graphing Equations Using Standard Form

11. Key Concepts, continued
Minimum
Maximum
y = 3x2
a = 3 and a > 0
y = –3x2
a = –3 and a < 0
5.3.1: Creating and Graphing Equations Using Standard Form

12. Key Concepts, continued
To create the equation of a quadratic in standard form, one must know some combination of the following key features: y-intercept, axis of symmetry or the vertex, and maximum or minimum.
The key features of a quadratic are the x-intercepts, y-intercept, where the function is increasing and decreasing, where the function is positive and negative, relative minimums and maximums, symmetries, and end behavior of the function.
5.3.1: Creating and Graphing Equations Using Standard Form

13. Key Concepts, continued
To create the equation of a quadratic given the vertex and y-intercept, first set the x-coordinate of the vertex equal to and isolate b.
Second, substitute the expression that is equal to b, the y-intercept, and the coordinates of the vertex into the standard form of a quadratic equation, y = ax2 + bx + c.
Solve the equation for a and then substitute the value of a into the equation you created in the first step to determine the value of b.
5.3.1: Creating and Graphing Equations Using Standard Form

14. Common Errors/Misconceptions
forgetting to write an equation for the axis of symmetry
not finding and labeling all important points
drawing the parabola as a straight line or a series of straight lines instead of a curve
5.3.1: Creating and Graphing Equations Using Standard Form

15. Guided Practice Example 2
h(x) = 2x2 – 11x + 5 is a quadratic function. Determine the direction in which the function opens, the vertex, the axis of symmetry, the x-intercept(s), and the y-intercept. Use this information to sketch the graph.
5.3.1: Creating and Graphing Equations Using Standard Form

16. Guided Practice: Example 2, continued
Determine whether the graph opens up or down.
h(x) = 2x2 – 11x + 5 is in standard form; therefore, a = 2.
Since a > 0, the parabola opens up.
5.3.1: Creating and Graphing Equations Using Standard Form

17. Guided Practice: Example 2, continued
Find the vertex and the equation of the axis of symmetry.
h(x) = 2x2 – 11x + 5 is in standard form; therefore,
a = 2 and b = –11.
The vertex has an x-value of 2.75.
Equation to determine the vertex
Substitute 2 for a and –11 for b.
x = 2.75
Simplify.
5.3.1: Creating and Graphing Equations Using Standard Form

18. Guided Practice: Example 2, continued
Since the input value is 2.75, find the output value by evaluating the function for h(2.75).
The y-value of the vertex is –
The vertex is the point (2.75, –10.125).
Since the axis of symmetry is the vertical line through the vertex, the equation of the axis of symmetry is x = 2.75.
h(x) = 2x2 – 11x + 5
Original equation
h(2.75) = 2(2.75)2 – 11(2.75) + 5
Substitute 2.75 for x.
h(2.75) = –10.125
Simplify.
5.3.1: Creating and Graphing Equations Using Standard Form

19. Guided Practice: Example 2, continued Find the y-intercept.
h(x) = 2x2 – 11x + 5 is in standard form, so the y-intercept is the constant c, which is 5.
The y-intercept is (0, 5).
5.3.1: Creating and Graphing Equations Using Standard Form

20. Guided Practice: Example 2, continued
Find the x-intercepts, if any exist.
The x-intercepts occur when y = 0.
Substitute 0 for the output, h(x), and solve.
This equation is factorable, but if we cannot easily identify the factors, the quadratic formula always works.
Note both methods shown on the following slides.
5.3.1: Creating and Graphing Equations Using Standard Form

21. Guided Practice: Example 2, continued
Solved by factoring:
h(x) = 2x2 – 11x + 5
0 = 2x2 – 11x + 5
0 = (2x – 1)(x – 5)
0 = 2x – 1 or 0 = x – 5
x = 0.5 or x = 5
Solved using the quadratic formula:
a = 2, b = –11, and c = 5
5.3.1: Creating and Graphing Equations Using Standard Form

22. Guided Practice: Example 2, continued
5.3.1: Creating and Graphing Equations Using Standard Form

23. Guided Practice: Example 2, continued
The x-intercepts are (0.5, 0) and (5, 0).
5.3.1: Creating and Graphing Equations Using Standard Form

24. Guided Practice: Example 2, continued
Plot the points from steps 2–4 and their symmetric points over the axis of symmetry.
Connect the points with a smooth curve.
5.3.1: Creating and Graphing Equations Using Standard Form

25. Guided Practice: Example 2, continued✔
5.3.1: Creating and Graphing Equations Using Standard Form

26. Guided Practice: Example 2, continued
5.3.1: Creating and Graphing Equations Using Standard Form

27. Guided Practice Example 5
Create the equation of a quadratic function given a vertex of (2, –4) and a y-intercept of 4.
5.3.1: Creating and Graphing Equations Using Standard Form

28. Guided Practice: Example 5, continued
Write an equation for b in terms of a.
Set the x-coordinate of the vertex equal to
Substitute 2 for x.
4a = –b
Multiply both sides by 2a.
–4a = b
Multiply both sides by –1.
b = –4a
5.3.1: Creating and Graphing Equations Using Standard Form

29. Guided Practice: Example 5, continued
Substitute the expression for b from step 1, the coordinates of the vertex, and the y-intercept into the standard form of a quadratic equation.
y = ax2 + bx + c
Standard form of a quadratic equation
y = ax2 + (–4a)x + c
Substitute –4a for b.
(–4) = a(2)2 + (–4a)(2) + c
Substitute the vertex (2, –4) for x and y.
5.3.1: Creating and Graphing Equations Using Standard Form

30. Guided Practice: Example 5, continued
(–4) = a(2)2 + (–4a)(2) + 4
Substitute the y-intercept of 4 for c.
–4 = 4a – 8a + 4
Simplify, then solve for a.
–8 = –4a
a = 2
5.3.1: Creating and Graphing Equations Using Standard Form

31. Guided Practice: Example 5, continued
Substitute the value of a into the equation for b from step 1.
b = –4a
Equation from step 1
b = –4(2)
Substitute 2 for a.
b = –8
Simplify.
5.3.1: Creating and Graphing Equations Using Standard Form

32. ✔ Guided Practice: Example 5, continued
Substitute a, b, and c into the standard form of a quadratic equation.
The equation of the quadratic function
with a vertex of (2, –4) and a y-intercept
of 4 is y = 2x2 – 8x + 4.
y = ax2 + bx + c
Standard form of a quadratic equation
y = 2x2 – 8x + 4
Substitute 2 for a, –8 for b, and 4 for c. ✔
5.3.1: Creating and Graphing Equations Using Standard Form

33. Guided Practice: Example 5, continued
5.3.1: Creating and Graphing Equations Using Standard Form