Quadratic Equations GRAPHING MAXMIN SOLVING Calculator Tips Main

Quadratic Equations * GRAPHING * MAX/MIN * SOLVING Calculator Tips Main Menu Back to last slide

MAIN MENU Introduction to Quadratic Equations Examples of Quadratic Equations Quadratic Equation Facts Quadratic Equation Practice Problems Calculator Tips Main Menu Back to last slide

INTRODUCTION TO QUADRATIC EQUATIONS Quadratic Equations can be written as follows: Standard Form OR Function Notation OR Vertex Form Calculator Tips Main Menu Back to last slide

Quadratics: A quadratic function always has a degree of 2. § That means that there’s always an x 2 in the equation and never any higher power of x. § There may or may not be a bx term and there may or may not be a c. NOTE: Quadratic equations never have more than one independent variable in the equation. The variable y is dependent, and f(x) means that the equation is a function of the independent variable x. Calculator Tips Main Menu Back to last slide

EXAMPLES OF QUADRATIC EQUATIONS Calculator Tips Main Menu Back to last slide

QUADRATIC EQUATION FACTS * The coefficient a in front of the x 2 term tells you if the graph opens up or down. • If a is positive, the graph opens up. • If a is a negative number, the graph opens down. * The a also gives you an idea if the graph is narrow or wide. • If a is a fraction, the graph is wide. • If it’s a number bigger than 1 or less than – 1, the graph is narrow. ) Calculator Tips Main Menu Back to last slide

Quadratic Equation Facts * The point where the graph reaches its highest point or its lowest point is called the vertex. - This point is an ordered pair. * The vertex can be either a maximum or a minimum. - If the graph opens up, The function has a minimum. Calculator Tips - If it opens down, the function has a maximum. Main Menu Back to last slide

Quadratic Equation Facts * Quadratic functions are always symmetric about a vertical line. * This line is called the axis of symmetry. * The vertex is always on the axis of symmetry, and this line acts like a mirror reflecting the graph. Vertex Calculator Tips Axis of symmetry Main Menu Back to last slide

Quadratic Equation Facts * The roots, solutions, zeroes, or x-intercepts of a quadratic equation all mean the same thing. * This is where the graph of the function crosses the x-axis. The y-value is always zero for Roots, these points. solutions, zeroes, x-intercepts Calculator Tips Main Menu Back to last slide

Quadratic Equation Practice Problems Calculator Tips Main Menu Back to last slide

• Type this equation into the Y 1= in your calculator. • Graph it. Adjust your window if needed. • Look at the graph. Find the top or the bottom of the graph—the “vertex. ” Is it a maximum or a minimum? Calculator Tips Main Menu Back to last slide

• Since this is the absolute lowest point that this graph can go, it is a minimum. Use your calculator to find the vertex, the axis of symmetry, and the minimum value of the function. Calculator Tips Vertex: (1. 5, -0. 25)a Axis of Symmetry: Minimum value of the function: y = -0. 25 Main Menu Back to last slide

• Now, find the solutions of this quadratic function on your calculator. (Remember that the solutions may also be called roots, zeroes, or xintercepts. ) • Choose an x-value to the left of this root (say 0, for example) for your Left Bound. • Choose an x-value to the right of this root (say 1. 5, for example) for your Right Bound. • Press “Enter” to Guess. Calculator Tips Solution: x=1 • Repeat this process for the other solution. You end up with x = 2. Main Menu Back to last slide

• Type this equation into the Y 1= in your calculator. • Graph it. Adjust your window if needed. • Look at the graph. Find the top or the bottom of the graph—the “vertex. ” Is it a maximum or a minimum? • Since this is the absolute highest point that this graph can go, it is a maximum. Use your calculator to find the vertex, the axis of symmetry, and the maximum value of the function. Calculator Tips Main Menu Vertex: (3, 2. 5) Axis of Symmetry: Maximum value of the function: y = 2. 5 Back to last slide

• Now, find the solutions of this quadratic function on your calculator. (Remember that the solutions may also be called roots, zeroes, or xintercepts. ) • Choose an x-value to the left of this root (say 0, for example) for your Left Bound. • Choose an x-value to the right of this root (say 1. 5, for example) for your Right Bound. • Press “Enter” to Guess. Calculator Tips Solution: x=0. 734 • Repeat this process for the other solution. You end up with x = 5. 24. Main Menu Back to last slide

• Type this equation into the Y 1= in your calculator. • Graph it. Adjust your window if needed. • Look at the graph. Find the top or the bottom of the graph—the “vertex. ” Is it a maximum or a minimum? • Since this is the absolute highest point that this graph can go, it is a maximum. Use your calculator to find the vertex, the axis of symmetry, and the maximum value of the function. Calculator Tips Main Menu Vertex: (0, 3) Axis of Symmetry: Maximum value of the function: y = 3 Back to last slide

• Now, find the solutions of this quadratic function on your calculator. (Remember that the solutions may also be called roots, zeroes, or xintercepts. ) • Choose an x-value to the left of this root (say -4, for example) for your Left Bound. • Choose an x-value to the right of this root (say -2, for example) for your Right Bound. • Press “Enter” to Guess. Calculator Tips Solution: x=-3. 46 • Repeat this process for the other solution. You end up with x = 3. 46. Main Menu Back to last slide

• Type this equation into the Y 1= in your calculator. • Graph it. Adjust your window if needed. • Look at the graph. Find the top or the bottom of the graph—the “vertex. ” Is it a maximum or a minimum? • Since this is the absolute lowest point that this graph can go, it is a minimum. Use your calculator to find the vertex, the axis of symmetry, and the minimum value of the function. Calculator Tips Main Menu Vertex: (0. 33, 2. 33) Axis of Symmetry: Minimum value of the function: y = 2. 33 Back to last slide

• Now, find the solutions of this quadratic function on your calculator. (Remember that the solutions may also be called roots, zeroes, or xintercepts. ) Since the graph of the function does not cross the x-axis, there are ? ? NO REAL SOLUTIONS. Calculator Tips Main Menu Back to last slide

Calculator Tips (Page 1) Entering and graphing the quadratic equation • Type your quadratic equation into Y 1= • Graph the equation. Press “Graph. ” • Adjust the window to see the vertex and the roots of the equation. • Use the Zoom Standard (6) button or the Zoom Fit (0) button to see more of the graph. You may also press “Window” and change the Xmin, Xmax, Ymin, and Ymax to see the graph. Calculator Tips Main Menu Back to last slide

Calculator Tips (Page 2) Finding the minimum value of a function * If the graph opens up, find the minimum value. – Press 2 nd, Trace (Calc), Minimum (3) – Move the cursor (using the arrow keys) to the left of the minimum. Press “Enter” for the “Left Bound” when you get to the point you want. – Move the cursor (using the arrow keys) to the right of the minimum. Press “Enter” for the “Right Bound” when you get to the point you want. – Press “Enter” again for the calculator to “Guess. ” – The x and y values that appear at the bottom of the graph are the x and y values of the vertex (x, y). The minimum value is the y value at this point. Main Menu Back to last slide

Calculator Tips (Page 3) Finding the maximum value of a function * If the graph opens down, find the maximum value. - Press 2 nd, Trace (Calc), Maximum (4) - Move the cursor (using the arrow keys) to the left of the maximum. Press “Enter” for the “Left Bound” when you get to the point you want. - Move the cursor (using the arrow keys) to the right of the maximum. Press “Enter” for the “Right Bound” when you get to the point you want. - Press “Enter” again for the calculator to “Guess. ” - The x and y values that appear at the bottom of the graph Tips are the x and y values of the vertex (x, y). Calculator Main Menu Back to last slide The maximum value is the y value at this point.

Calculator Tips (Page 4) Finding the roots or solutions of a quadratic • To find the roots, solutions, equation zeroes, or x-intercepts of the equation • Make sure you can see the root(s) in the viewing window. If not, adjust your window. • Press 2 nd, Trace (Calc), Zero (2) • Select an x-value to the left of the root you want to find. Press “Enter” for the “Left Bound. ” • Select an x-value to the right of the root you want to find. Press “Enter” for the “Right Bound. ” • Press “Enter” again to guess. • The x-value that appears in the window is the root. The yvalue should be 0 (or very Main close Menu to 0). Back to last slide

CALCULATOR TIPS (PAGE 5) Finding the axis of symmetry of a quadratic equation • Find the maximum or the minimum using the instructions on the preceding pages. The axis of symmetry is the equation for x = that appears on the screen for the maximum or the minimum. Calculator Tips Main Menu Back to last slide