Quadratics 1


Introduction to Quadratics

Quadratic: relating to an equation with an exponent of 2

Quadratics can be applied to many real-life applications, such as business plans for revenue

It is the foundation for Grade 10 and 11 mathematics

Characteristics of a Quadratic Relation:

  • When written in vertex form, has the exponent 2 applied to (x - h)
  • Its shape on a graph (a parabola) appears like a curve
  • It "curves" up or down to its vertex (highest or lowest point) before coming down or up

  • The second differences of a quadratic relationship, when written on a table, are the same
  • This is different from a linear relationship, where the first differences are the same
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Parts of a Parabola:

  • Parabola: the name of a quadratic shape on a graph
  • Vertex: the highest or lowest y-value on a quadratic graph, and its corresponding x-value
  • Axis of Symmetry: the vertex's x-value
  • Optimal Value: the vertex's y-value
  • Direction of Opening: (up or down:) the direction in which the parabola opens
  • Zeros: are the x-intercepts of a quadratic graph
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Vertex Form:

  • Written in the form: y = a (x - h)^2 + k

  • h determines the x-value of the parabola's vertex
  • However, h's positive or negative number will become opposite on the x-axis (if h is a positive number, it will become negative, and vice-versa)

  • k determines the y-value of the parabola's vertex
  • "k"'s number directly translates to a y-value

  • Using vertex form, you should be able to find y-intercepts, x-intercepts, the vertex, and other points
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Factored Form

  • Written in the form: y = a (x - s) (x - t)
  • From it, you can find the x-intercepts of the parabola, as well as the vertex

  • You must substitute x with individual values that cancel out with either - s or - t, depending on which bracket each x is in
  • This will give you the parabola's intercepts

  • Then, add or subtract your x-values, and divide the result by two
  • This will give you the vertex's x value

  • To find the vertex's y-value, substitute the x variables in the factored equation with the vertex's x-value, that you just got


  • You can also find the y-intercept of the parabola using factored form
  • Just substitute the x-variable with the number 0
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Step Pattern

  • Parabolas increase at x any y values exponentially
  • A base parabola (y = x) ^2 will increase at the following x and y-values:

1, 1

2, 4

3, 9

And so on...

  • However, if a value on a vertex form for "a" is a value other than 1, this step pattern will be somewhat different
  • The x-values will remain the same, but the y-values for each step will be multiplied by a's value
  • This is otherwise known as the factor that the equation is multiplied by

  • Remember, the step pattern extends from the vertex, and extends symmetrically on both sides of it
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Factors' Impact on a Parabola

  • A factor that is higher than 1 will vertically stretch the parabola
  • A factor that is lower than 1 will vertically compress the parabola
  • A factor that is negative will flip the parabola in the x-axis

The Impact of the h-Value on a Parabola

  • On a vertex form of a quadratic relation, the h-value can horizontally shift the entire parabola
  • This depends on whether the h-value is negative or positive
  • Remember that negative values translate to opposite positive ones, and vice-versa

Types of Word Problems

Minimizing Costs and Maximizing Revenue

The revenue. R, from T-shirt sales at an event is calculated as (number of shirts sold) x (price of each shirt). The current price of a T-shirt is $16, and 50 T-shirts are typically sold. For each $2 increase in the price of a T-shirt, five fewer T-shirts are sold.

a) Write this information in the form R = a(x - r) (x - s)

b) What price maximizes the revenue?

c) What is the maximum revenue?

Skills needed:

  • Understanding how to analyze information to place them into quadratic equations, such as into standard form
  • Finding x-intercepts from standard form equations, and using the intercepts to find a vertex
  • Understanding how to translate information out of answers you find

Finding Mathematical Relationships in the Real World

Shape Problems

Flight Situations

"The predicted flight path of a toy rocket used in a mathematics project is defined by the relation h = -3(d - 2) (d - 12), where d is the horizontal distance, in metres, from a wall, and h is the height, in metres, above the ground."

a) How far from the wall is the rocket when it is launched?

b) How far is the rocket from the wall when it lands on the ground?

c) What is the maximum height of the rocket, and how far, horizontally, is it from the wall at that moment?

Skills needed:

  • Finding x-intercepts from standard form equations, and using the intercepts to find a vertex
  • Understanding how to translate information out of answers you find

Find the Equation of a Relationship Given Key Information

Reflection on this Unit

This unit, I felt that I had an overall good understanding of the concepts that were taught in class. Since this was the first quadratics unit, the concepts were somewhat abstract to me at first, but i was able to make sense of what was taught and be able to visualize it in my head -- as I paid attention in class and completed all of the homework.

I especially did well with the different types of word problems that were taught, and I was able to use my skills of analyzing quadratic expressions and equations well to gather information from them, including finding the vertex and x-intercepts. This worked well with my ability to understand written word problems.

As well, I understood the workings of vertex form quadratic equations and factored form equations well, and was able to utilize them during the unit. However, I struggled a little with understanding the individual parts of each type of equation, and how they differed from each other. To support my knowledge, I could have re-written the information that I was given in the unit multiple times, and I could have made my own personalized notes.

Overall, I did well in this unit; however, my final unit test was a bit lower than what I had expected. Next time I will aim to do better.

Unit Test from Quadratics 1

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