Quadratic Functions (a) table of values (b) graph (c)equation.pptx

Math 9

1. MATHEMATICS 9 QUADRATIC FUNCTIONS Using… Table of values Graph equation

2. Quadratic functions can be analyzed and represented in three different ways: (a) using a table of values, (b) graphically, and (c) using the equation. Let's explore each method:

3. (a) Table of Values: To create a table of values for a quadratic function, choose different values for x and calculate the corresponding y-values using the quadratic equation. Here's an example:

4. Consider the quadratic function f(x) = 2x 2 - 3x + 1. Choose some x-values, say -2, -1, 0, 1, and 2. Calculate the corresponding y-values using the equation: f(x) = 2x 2 - 3x + 1.

5. Using the values selected, we can generate a table like this: x f(x) -2 11 -1 6 1 1 2 1

6. (b) Graph: To graph a quadratic function, plot points on a coordinate plane using a table of values or by recognizing important characteristics of the function. In this case, we can plot the points from the table or calculate additional points if needed. Then, connect the points to form a smooth curve.

7. For the quadratic function f(x) = 2x 2 - 3x + 1, the graph will be a parabola. The general shape and direction of the parabola can be determined by the coefficient "a" in the quadratic equation. In this case, since "a" is positive (a=2), the parabola opens upward.

8. (c) Equation: The quadratic function can also be represented using the equation. In this case, the equation is f(x) = 2x 2 - 3x + 1. The equation provides valuable information about the function, such as the coefficients (a, b, c) and key features like the vertex, axis of symmetry, and the direction of opening.

9. For example, using the quadratic formula, we can find the vertex of the parabola, which is given by: x = -b / (2a) = -(-3) / (2 * 2) = 3 / 4 To find the corresponding y-value, substitute x = 3 / 4 into the equation: f( 3 / 4 ) = 2( 3 / 4 ) 2 - 3( 3 / 4 ) + 1 = 1 / 8

10. So, the vertex is ( 3 / 4 , 1 / 8 ), and the axis of symmetry is x = 3 / 4 . By analyzing the equation, we can also determine the y-intercept, x-intercepts (if any), and the minimum or maximum value achieved by the function.

11. Remember, the equation, graph, and table of values collectively provide a comprehensive understanding of the quadratic function.

12. Sample Solving:

13. (a) Using table of Values Step 1: The equation of the quadratic function is f(x) = 2x^2 - 3x + 1. Step 2: Let's say we want to calculate the values for x ranging from -2 to 2. Step 3: Substitute each x-value into the equation to find the corresponding y-values: - For x = -2: f(-2) = 2(-2)^2 - 3(-2) + 1 = 12 - For x = -1: f(-1) = 2(-1)^2 - 3(-1) + 1 = 6 - For x = 0: f(0) = 2(0)^2 - 3(0) + 1 = 1 - For x = 1: f(1) = 2(1)^2 - 3(1) + 1 = 0 - For x = 2: f(2) = 2(2)^2 - 3(2) + 1 = 3 Step 4: Create a table with two columns: | x | f(x) | |-------|---------| | -2 | 12 | | -1 | 6 | | 0 | 1 | | 1 | 0 | | 2 | 3 |