Narration: Which expression did you find easiest to simplify? Why?
Image description: Three expressions are shown. The expression four squared minus the quantity ten times four plus sixteen, the quantity four minus two times the quantity four minus eight, and the quantity four minus five squared minus nine.
Narration: All of these expressions simplify to the same value, negative 8.
Image description: The expressions shift upward as equals negative 8 appears beneath each of them.
Narration: Each expression is written in a particular form, and each form has a distinct purpose. All of these expressions are connected in a very important way. Use the following directions to help you identify the connection between these three expressions. To begin, let's replace each 4 in the expressions with the variable x.
Image description: The previous expressions shift upward. The expression four squared minus the quantity ten times four plus sixteen becomes the expression x squared minus ten x plus sixteen. The quantity four minus two times the quantity four minus eight becomes the expression the quantity x minus two times the quantity x minus eight. The quantity four minus five squared minus nine equals negative eight becomes the expression the quantity x minus five squared minus nine.
Narration: This allows us to later solve for the variable y. What do you notice? Each expression is a quadratic expression in a different form. Consider each of these expressions as functional inputs.
Image description: Three expressions appear along with their graphs. The first expression is the equation y equals x squared minus ten x plus sixteen. The graph shows a parabola that opens up, intersects the x-axis at two and eight, and has a vertex at the point five comma negative nine. The second expression is the equation y equals the quantity x minus two times the quantity x minus eight. The graph shows a parabola that opens up, intersects the x-axis at two and eight, and has a vertex at the point five comma negative nine. The third expressions is the equation y equals the quantity x minus five squared minus nine. The graph shows a parabola that opens up, intersects the x-axis at two and eight, and has a vertex at the point five comma negative nine.
Narration: Setting each expression equal to y, we get quadratic functions that can be graphed on the coordinate plane. Notice the parabolas formed by these quadratic functions are all the same! This is because the quadratic expressions are equivalent expressions, forming equivalent functions when set equal to y.
Image description: Y equals appears before each expression.
Narration: Each form of the quadratic equation has a special purpose that relates to the parabola graphed. Begin with the function: y equals x squared minus ten x plus 16.
Image description: The equation y equals x squared minus ten x plus sixteen appears. The graph shows a parabola that opens up, intersects the x-axis at two and eight, and has a vertex at the point five comma negative nine. The standard form is y equals a times x squared plus b times x plus c.
Narration: This quadratic function is in standard form.
Image description: The words standard form appear onscreen.
Narration: The form is referred to as standard because it represents simplicity and uniformity that is universally accepted by mathematicians. It is how quadratic functions are most commonly written. Standard form of a quadratic function is written in the form: y equals ax squared plus bx plus c. The values of a, b and c will transform the shape and location of the parabola on the coordinate grid.
Image description: y equals ax squared plus bx plus c appears onscreen. The variables a, b and C are highlighted along with the shape of the parabola on the graph.
Narration: Now, look at the function written as: y equals the quantity x minus 2 times the quantity x minus 8.
Image description: the equations and words disappear, while the graph remains. An equation appears above the graph, y equals the quantity x minus 2 times the quantity x minus 8.
Narration: This quadratic function is in factored form.
Image description: The words factored form appear.
Narration: You can see that each term within parentheses represents one of two factors that can be multiplied together to obtain a product. Factored form of a quadratic function is written in the form: y equals a times the quantity x minus m times the quantity x minus n.
Image description: y equals a times the quantity x minus m times the quantity x minus n appears beneath the graph.
Narration: The value of a indicates a parabola's width.
Image description: The variable a is highlighted. An animation shows the parabola stretching and then contracting back to its original size and shape. The smaller the value of a, the more the parabola stretches open; the larger the value of a, the less the parabola opens. If the opening of the parabola is up, a is greater than zero.
Image description: An arrow points upward from the vertex of the parabola. A is greater than zero appears onscreen.
Narration: If the opening is down, a is less zero.
Image description: The parabola and arrow are flipped in the opposite direction and a is less than zero appears onscreen. Next, the parabola goes back to its original position.
Narration: The values of m and n are the zeros of the quadratic function. The zeros of a quadratic function are where the parabola crosses the x-axis.
Image description: zeroes colon m and n appears onscreen. Two dots appear at 2 and 8 on the x axis.
Narration: They are called zeros because they are the x values that cause y to equal zero.
Image description: The equation y equals a times the quantity x minus m time the quantity x minus n disappears from the screen as the graph is enlarged.
Narration: The zeros of the quadratic function y equals the quantity x minus 2 times the quantity x minus 8 are 2 and 8.
When x = 2, y = 0.
When x = 8, y = 0.
Note, a = 1.
Image description: The numbers 2 and 8 replace m and n in zeros colon m and n. Equations show when x = 2, y = 0 and when x = 8, y = 0. When x equals two, y equals the quantity x minus two times the quantity x minus eight results in y equals the quantity two minus two times the quantity two minus eight. This simplifies to y equals zero times negative six, which results in y equals zero. When x equals eight, y equals the quantity x minus two times the quantity x minus eight results in y equals the quantity eight minus two times the quantity eight minus eight. This simplifies to y equals six times zero, which results in y equals zero.
Animations show how the graph of y equals the quantity x minus 2 times the quantity x minus 8 can transform to the Parent Function y equals x squared.
Narration: The parabola is the same width as the parent quadratic function y equals x squared. Look at the standard form, y equals x squared minus ten x plus 16, and the factored form y equals the quantity x minus 2 times the quantity x minus 8.
Image description: Standard form is y equals x squared minus ten x plus sixteen. Factored form is y equals the quantity x minus two times the quantity x minus eight.
Narration: Earlier in the video, you saw how the graphs were the same. This means these equations must be equivalent. Using the distributive property, followed by combining like terms, you can see how the factored form is equivalent to the standard form.
Image description: The factored form is multiplied such that x is multiplied to x and the negative eight, and the negative two is multiplied to the x and the negative eight. This results in y equals x squared minus two x minus eight x plus sixteen. When simplified y equals x squared minus ten x plus sixteen.
Narration: Even though the standard form is the way most quadratic equations are written, the factored form is important because it emphasizes the zeros of the parabola.
Image description: y equals x squared minus ten x plus sixteen animates and slides onscreen for emphasis. The words standard form appear above the equation. Next the words factored form appear above the equation y equals a times the quantity x minus m times the quantity x minus n. The graph appears showing a parabola that opens up, intersects the x-axis at two and eight, and has a vertex at the point five comma negative nine.
Narration: Lastly, look at the function written as y equals the quantity x minus 5 squared minus 9. This quadratic function is in vertex form.
Image description: y equals the quantity x minus 5 squared minus 9 appears above the graph. This equation is replaced with y equals a times the quantity x minus h squared plus k.
Vertex form of a quadratic function is written in the form: y equals a times the quantity x minus h squared plus k.
Narration: Remember, the value of a stretches the parabola for smaller values of a, and shrinks the parabola for larger values of a. If the parabola opens up, a is greater than zero. If it opens down, a is less than zero.
Image description: An arrow points up from the vertex as a is greater than 0 appears, and then the parabola is flipped and an arrow points downward, as a is less than 0 appears.
Narration: The values of h and k represent the coordinates for the vertex, (h, k) of the quadratic function. The vertex of this quadratic function is point 5, negative 9. Note, a = 1.
Image description: The vertex 5, negative nine appears. Then the equations animate and 1 replaces a in the equation. Two graphs appear with equations above them. The first equation is y equals the quantity x minus five squared minus nine. The graph shows the corresponding parabola, which opens up, has zeros at two and eight, and a vertex at the point five negative nine. This equation and graph are labeled Vertex Form. The next equation and graph, labeled Parent Function show the y equals x squared above its graph.
Narration: The parabola is the same width as the parent quadratic function y equals x squared.
Look at the standard form, y equals x squared minus ten x plus 16, and the vertex form y equals the quantity x minus 5 squared minus 9.
Image description: The standard and vertex forms appear.
Narration: Remember, on a previous slide you saw how the graphs were the same. This means these equations must be equivalent. Using the distributive property, followed by combining like terms, you can see how the vertex form is equivalent to the standard form. The vertex form is important because it emphasizes the vertex of the parabola.
Image description: Standard form is y equals x squared minus ten x plus sixteen. Vertex form is y equals the quantity x minus five squared minus nine. The function y equals the quantity x minus five squared minus nine is multiplied such that y equals the quantity x minus five times the quantity x minus five minus nine. The x is multiplied to the x and the negative five, and the negative five is multiplied to the x and the negative five, producing y equals x squared minus five x minus five x plus twenty-five minus nine. This simplifies y equals x squared minus ten x plus sixteen. Vertex form and the graph representing the parabola appears.