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| 17 Theory slides |
| 8 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
All three graphs of the functions presented earlier have a very distinctive form. These curves have a specific name, which now will be properly introduced.
A parabola can be vertical or horizontal. A vertical parabola can open upward or downward. In comparison, a horizontal parabola can open to the left or the right.
Equations of parabolas always contain a variable raised to the second power. This is why the functions that represent vertical parabolas are called quadratic functions.
Name | Equation | Characteristics |
---|---|---|
Standard Form | y=ax^2+bx+c | a, b, and c are real numbers, a≠ 0, and c is the y-intercept of the parabola. |
Vertex Form | y=a(x-h)^2+k | a, h, and k are real numbers, a≠ 0, and (h,k) is the vertex of the parabola. |
Intercept Form (also called Factored Form) |
y=a(x-x_1)(x-x_2) | a, x_1, and x_2 are real numbers, a≠ 0, and x_1 and x_2 are the x-intercepts of the parabola. |
The inherent shape of parabolas gives rise to several characteristics that all quadratic functions have in common.
A parabola either opens upward or downward. This is the direction of the parabola . If the leading coefficient a of the corresponding equation is positive, the parabola opens upward. If the coefficient is negative, the parabola opens downward.
What is more, if a function's axis of symmetry is the y-axis, it is an even function.
The concept of an axis of symmetry extends beyond graphs of functions. Any geometric figure can have an axis of symmetry if a line exists that divides the figure into congruent, mirror-image halves.
A parabola can intersect the x-axis at zero, one, or two points. Since the function's value at an x-intercept is always 0, these points are called zeros, or sometimes roots.
Because all graphs of quadratic functions extend infinitely to the left and right, they each have a y-intercept somewhere along the y-axis.
Consider the given parabola. Identify its zeros, line of symmetry, or y-intercept.
A quadratic function is said to be written in vertex form if it has the following format.
y=a(x-h)^2+k
Here, a, h, and k are real numbers with a≠ 0. The value of a gives the direction of the parabola. When a > 0, the parabola faces upward, and when a < 0, it faces downward. The vertex of the parabola lies at (h,k), and the axis of symmetry is the vertical line x=h. Consider the graph of y=- 14(x-4)^2+8.
Comparing the generic vertex form with the example function, the values of a, h, and k can be identified. Vertex Form:& y= a(x- h)^2+ k Example Function:& y= - 1/4(x- 4)^2+ 8 These values determine the characteristics of the parabola shown in the graph.
Direction | Vertex | Axis of Symmetry |
---|---|---|
a= - 1/4 | h= 4 and k= 8 | h= 4 |
Since - 14 is less than 0, the parabola opens downward. | The vertex is located at ( 4, 8). | The axis of symmetry is the vertical line x= 4. |
Consider other example quadratic functions. Function1:& y=2(x+1)^2+7 Function2:& y= (x-3)^2+1 Function3:& y=5(x-2)^2-3 Although these functions do not strictly follow the format for the vertex form, they are said to be written in vertex form because they can easily be rewritten in the desired format.
Function 1 | Function 2 | Function 3 |
---|---|---|
y=2(x+1)^2+7 ⇕ y= 2(x-( - 1))^2+ 7 |
y=(x-3)^2+1 ⇕ y= 1(x- 3)^2+ 1 |
y=5(x-2)^2-3 ⇕ y= 5(x- 2)^2+( - 3) |
When a quadratic function is written in vertex form, some characteristics of its graph can be identified.
y=a(x-h)^2+k | ||
---|---|---|
Direction | Vertex | Axis of Symmetry |
a>0 &⇒ upward a<0 &⇒ downward | (h,k) | x=h |
To begin, identify the vertex (h,k) from the function rule. y=(x-2)^2-4 ⇕ y=(x- 2)^2+( - 4) Here, h= 2 and k= - 4. Therefore, the vertex of the parabola is ( 2, - 4). This point can be plotted on a coordinate plane.
The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two mirror images. Since the axis of symmetry is x=h, here it is x= 2.
Now, with three points plotted, the direction of the parabola can be seen. It appears that the parabola faces upward. Identify the value of a in the equation to see if this is correct. y=(x-2)^2-4 ⇕ y= 1(x-2)^2-4 Since a= 1, which is greater than zero, it can be said that the parabola opens upward. To graph the quadratic function, connect the three points with a smooth curve.
LaShay loves playing golf.
To identify the vertex of the parabola, the given equation can be compared to the general form of a quadratic function written in vertex form. General Form y= a(x- h)^2+ k [1.2em] Given Function y= - 2(x- 3/2)^2+ 9/2 For the given function, h= 32 and k= 92. This means that the vertex of the parabola is ( 32, 92). Since the parabola illustrates the position of the ball, the first quadrant will only be considered.
The axis of symmetry of the parabola is the vertical line through the vertex. Therefore, in this case, the equation of the axis of symmetry is x= 32.
x= 0
Subtract term
(- a)^2 = a^2
(a/b)^m=a^m/b^m
a*b/c= a* b/c
a/b=.a /2./.b /2.
Add fractions
Mark is studying parabolas so that he can help LaShay with her golf swing. He wants to write the vertex form of the quadratic function that corresponds to the given graph.
Start by identifying the vertex of the parabola.
Recall the format of the vertex form. y= a(x- h)^2+ k In this format, the vertex of the parabola has coordinates ( h, k). Therefore, to state the values of h and k, start by identifying the vertex.
The vertex is the point with coordinates ( 3, - 2). This means that in the equation that corresponds to the given parabola, h= 3 and k= - 2. y= a(x- 3)^2+( - 2) ⇕ y= a(x-3)^2-2 Finally, to find the value of a, any point on the parabola can be used. For simplicity, the y-intercept will be used.
x= 0, y= 1
Subtract term
(- a)^2 = a^2
LHS+2=RHS+2
.LHS /9.=.RHS /9.
a/b=.a /3./.b /3.
Rearrange equation
A quadratic function is said to be written in factored form, or intercept form, if it follows a specific format.
y=a(x-p)(x-q)
Here, a, p, and q are real numbers with a≠ 0. The value of a gives the direction of the parabola. When a> 0, the parabola faces upward, and when a< 0, it faces downward. The zeros of the parabola are p and q, and the axis of symmetry is the vertical line x= p+q2.
Consider the graph of y= 12(x-7)(x-13).
Comparing the generic factored form with the example function, the values of a, p, and q can be identified. Factored Form:& y= a(x- p)(x- q) Example Function:& y= 1/2(x- 7)(x- 13) These values determine the characteristics of the parabola shown in the graph.
Direction | Zeros | Axis of Symmetry |
---|---|---|
a= 1/2 | p= 7 and q= 13 | p+ q/2 ⇓ 7+ 13/2= 10 |
Since 12 is greater than 0, the parabola opens upward. | The zeros are 7 and 13. Therefore, the parabola intersects the x-axis at ( 7,0) and ( 13,0). | The axis of symmetry is the vertical line x= 10. |
Consider other example quadratic functions. Function1:& y=2(x+1)(x-3) Function2:& y=(x-5)(x-9) Function3:& y=5x(x-2) Although these functions do not strictly follow the format of the factored form, they are said to be written in factored form because they can easily be rewritten in the desired format.
Function 1 | Function 2 | Function 3 |
---|---|---|
y=2(x+1)(x-3) ⇕ y= 2(x-( - 1))(x- 3) |
y=(x-5)(x-9) ⇕ y= 1(x- 5)(x- 9) |
y=5x(x-2) ⇕ y= 5(x- 0)(x- 2) |
In the following applet, several quadratic functions are expressed in different forms. Are the quadratic functions written in vertex form, factored form, or neither?
When a quadratic function is written in factored form, some characteristics of its graph can be identified.
y=a(x-p)(x-q) | ||
---|---|---|
Direction | Zeros | Axis of Symmetry |
a>0 &⇒ upward a<0 &⇒ downward | p and q | x=p+q/2 |
It is possible to graph a quadratic function using these characteristics. Consider the function y=-(x+1)(x-5).
To begin, identify the zeros p and q from the function rule. y=- (x+1)(x-5) ⇕ y=- (x-( - 1))(x- 5) Here, p= - 1 and q= 5. Therefore, the x-intercepts of the parabola are ( - 1,0) and ( 5,0). These points can be plotted on a coordinate plane.
x= 2
Add and subtract terms
Remove parentheses
- a(- b)=a* b
Now, with the three points plotted, the direction of the parabola can be seen. It appears that the parabola faces downward. Identify the value of a in the equation to see if this is correct. y=- (x+1)(x-5) ⇕ y= - 1(x+1)(x-5) Since a= - 1, which is less than zero, it can be said that the parabola opens downward. To graph the quadratic function, connect the three points with a smooth curve.
By now, it is not a secret that LaShay loves playing golf.
To identify the zeros of the quadratic function, the given equation can be rewritten and compared to the general form of a quadratic function written in factored form. General Form y= a(x- p)(x- q) [1.2em] Given Function y=- 2x(x-4) ⇕ y= - 2(x- 0)(x- 4) For the given function, p= 0 and q= 4. This means that the zeros of the function are 0 and 4, so the parabola intersects the x-axis at ( 0,0) and ( 4,0).
Note that, for the given function, the value of a is - 2. Therefore, the parabola opens downward. This corresponds to the loci of the points plotted in the coordinate plane. Finally, these points can be connected with a smooth curve to draw the parabola.
Continuing his studies, Mark wants to write the factored form of the quadratic function that corresponds to the given parabola.
Start by identifying the x-intercepts of the parabola.
Recall the format of the factored form of a parabola. y= a(x- p)(x- q) In this format, the zeros of the function are p and q. Therefore, to state the values of p and q, start by identifying the x-intercepts of the graph.
The parabola intersects the x-axis at ( - 5,0) and ( 1,0). This means that p and q are - 5 and 1. It does not matter which value is attributed to which variable, so it will arbitrarily be considered that p= - 5 and q= 1. With this information, the following equation can be written. y= a(x-( - 5))(x- 1) ⇕ y= a(x+5)(x-1) Finally, to find the value of a, any point on the parabola can be used. For simplicity, the vertex will be used.
x= - 2, y= - 3
Add and subtract terms
a(- b)=- a * b
.LHS /(- 9).=.RHS /(- 9).
- a/- b=a/b
a/b=.a /3./.b /3.
Rearrange equation
For the following quadratic functions, identify the vertex or the zeros.
(a-b)^2=a^2-2ab+b^2
Commutative Property of Multiplication
Multiply
Calculate power
Distribute - 2
Distribute (x-3)
Distribute - 2x & 2
Add terms
Let's begin by reviewing the intercept form of a quadratic function. f(x)=a(x-p)(x-q) In this form, a, p, and q are real numbers with a≠ 0. The value of a gives the direction of the corresponding parabola, whether it opens upward or downward. The x-intercepts of the function are p and q. To find the x-intercepts of the given function, which is written in standard form, we can try to rewrite it in its intercept form.
Standard Form | Intercept Form |
---|---|
f(x)=kx^2-k^3 | ? |
To do so, start by factoring out k. Then, factor out the resulting difference of squares.
The function is now written in its intercept form. f(x)=k(x+k)(x-k) ⇕ f(x)=k(x-(- k))(x-k) Finally, we can conclude that the x-intercepts of the function are x=- k and x=k.
Consider the following quadratic function. f(x)=a(x-p)(x-q)
Let's begin by reviewing the intercept form of a quadratic function. f(x)=a(x-p)(x-q) In this form, p and q are the zeros of the function and a is a real number other than 0. We are told that the parabola passes through the origin. This means that one of the zeros is 0. p=0 or q=0 In either case, the product is zero by the Zero Property of Multiplication. Therefore, we can conclude that pq=0.
We are given that the product of p and q is positive. Start by noticing that, based on the information we found in Part A, neither p nor q is equal to 0. pq > 0 ⇓ p ≠ 0 and q ≠ 0 Next, we can use the fact that if the product of two numbers is positive, both numbers must be positive or both numbers are negative.
Case | Example | Product | Is Product Positive? |
---|---|---|---|
Both positive | 2 and 6 | 2* 6 = 12 | ✓ |
Both negative | -6 and -3 | -6 * (-3) = 18 | ✓ |
One positive, one negative | -5 and 3 | -5 * 3 = -15 | * |
This means that if pq is greater than 0, p and q must have the same sign.