Chapter 7 Quadratic Equations and Functions

TSWBAT solve quadratic equations by completing the square method.

Completing the Square Remember we have a number of types of equations: Linear – ax + b = 0 Quadratic – ax2 + bx + c = 0 Cubic – ax3 + bx2 + cx +d = 0 In this section we will work on all types of quadratic equations. We have already learned how to solve quadratic equations by factoring and by taking the square root. Now we will combine those ideas in this concept of Completing the Square.

Completing the Square From Chapter 5 we could solve a quadratic equation by factoring. When we factored Perfect Square Trinomials we got (x+y)2=0. Now combining that idea with the square root rules we just learned we can factor a PST equation and not just have it set equal to zero. It now can be set equal to any number. Example (2x+3y)2=5

Completing the Square This is fine as long as we have a PST. But if we don’t have a PST we use this process of Completing the Square to get a PST and other number. There are five steps to follow in this process. Step 1 – transform the equation so that the constant term c is alone on the right side. ax2 + bx + c = 0 -> ax2 + bx = - c

Completing the Square Step 2 – If a is not equal to 1, divide the whole equation by a. If it is move on to step 3. So a≠1 then Step 3 – take the second term (the bx term) and add the square of half the b to BOTH sides of the equation. (Completing the Square)

Completing the Square Step 4 – Factor the left side as the perfect square. Step 5 – Solve using the square root method.

Completing the Square Example – 2y2 + 2y + 5 = 0 Step 1 - 2y2 + 2y = - 5 Step 2 – Step 3 –

Completing the Square Step 4 – Step 5 –

TSWBAT solve quadratic equations by the quadratic formula.

Quadratic Formula The Quadratic Formula is another method to solve a non-factorable quadratic equation but does not get as complex as completing the square method does. The formula is where a, b, and c come from the coefficients of the quadratic equation ax2+bx+c=0.

Quadratic Formula Example: Solve 3x2+x-1=0 So a=3, b=1, and c=-1

Discriminant TSWBAT determine the nature of the roots of a quadratic equation using its discriminant.

Discriminant The discriminant comes from the quadratic formula and gives a person insight into the roots of a quadratic equation. The discriminant is the term under the square root symbol. Example Where D is the discriminant.

Discriminant The discriminant tells us that the roots are one of three options. If the Discriminant is Positive - the roots are real and unequal. (Usually they will be conjugates). Example D=11 X= If the Discriminant is Zero – the roots are real and equal. (Double Root). Example D=0 X= If the Discriminant is negative – the roots are imaginary and unequal. (Imaginary Conjugates Example D= - 4 X=

Discriminant Test for Rational Roots – if the quadratic equation has coefficients that are integers and its discriminant is a perfect square, then the equation has rational roots. Also if an equation can be transformed into an equivalent equation that meets this test, then it has rational roots. Example

Quadratic Form Equations TSWBAT recognize and solve equations in quadratic form.

Equations in Quadratic Form You can also use the Quadratic Formula to solve equations with the variable as part of a function. We can have 3x-2 as our variable term or 1/2x. In general an Equation in Quadratic Form is an equation that has a function of x in place of the variable. a(f(x))2+b(f(x))+c=0

Equations in Quadratic Form Example:

Graphing Parabolas TSWBAT

Graphing Parabolas Example – See board or Overhead. Parabola – the graph of the equation y=x2. Axis of Symmetry – the vertical line along which the graph of a parabola can be folded and yield the mirror of one side on the other side. Vertex – the point of a parabola where the parabola crosses it’s axis of symmetry. This is also either the highest or lowest point on the graph of a parabola. Example – See board or Overhead.

Graphing Parabolas We can also move a parabola along the x-axis by adding a number h to the equation making it y=(x-h)2. Example: – See board or Overhead. We can also move a parabola along the y-axis by adding a number k to the equation making it y-k=(x-h)2. We can also adjust the sharpness of the curve of the parabola by adding a number a to the equation making it y-k=a(x-h)2.

Graphing Parabolas Along with all of this we can also find the x and y intercepts of the graph of a parabola. X-Intercept(s) – the point(s) where the parabola crosses the x-axis. Y-Intercept(s) – the point(s) where the parabola crosses the y-axis. Example:

Graphing Parabolas Finally we can write the equation of a parabola if we know the vertex and another point along the parabola. Example: Find the equation of the parabola having vertex (1,-2) and containing the point (3,6).

Quadratic Functions Quadratic Function – a function that can be written in either of two forms. General Form: f(x) = ax2 + bx + c (a≠0) Completed-Square form: f(x) = a(x-h)2 + k (a≠0) Regardless of the form of the function, the graph of any quadratic function is a parabola.

Quadratic Functions From the general form of the function we also receive some insight about the parabola. So letting f(x) = ax2 + bx + c (a≠0) If a<0, f has a maximum value for a vertex. If a>0, f has a minimum value for a vertex. This maximum or minimum value of f is the y-coordinate when x= - b/2a, which is the vertex. Graphing the parabola from the function: Example 1 - Example 2 -

Writing Quadratic Equations and Functions

Writing Quadratic Equations and Functions