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Course: Digital SAT Math > Unit 4
Lesson 11: Quadratic graphs: foundationsQuadratic graphs | Lesson
A guide to quadratic graphs on the digital SAT
What are quadratic functions?
In a quadratic function, the of the function is based on an expression in which the is the highest power term. For example, is a quadratic function, because in the highest power term, the is raised to the second power.
Unlike the graphs of linear functions, the graphs of quadratic functions are nonlinear: they don't look like straight lines. Specifically, the graphs of quadratic functions are called parabolas.
In this lesson, we'll learn to:
- Graph quadratic functions
- Identify the features of quadratic functions
- Rewrite quadratic functions to showcase specific features of graphs
- Transform quadratic functions
You can learn anything. Let's do this!
How do I graph parabolas, and what are their features?
Parabolas intro
What are the features of a parabola?
All parabolas have a -intercept, a , and open either upward or downward.
Since the vertex is the point at which a parabola changes from increasing to decreasing or vice versa, it is also either the maximum or minimum -value of the parabola.
- If the parabola opens upward, then the vertex is the lowest point on the parabola.
- If the parabola opens downward, then the vertex is the highest point on the parabola.
A parabola can also have zero, one, or two -intercepts.
Note: the terms "zero" and "root" are used interchangeably with " -intercept". They all mean the same thing!
Parabolas also have vertical symmetry along a vertical line that passes through the vertex.
For example, if a parabola has a vertex at , then the parabola has the same -values at and , at and , and so on.
To graph a quadratic function:
- Evaluate the function at several different values of
. - Plot the input-output pairs as points in the
-plane. - Sketch a parabola that passes through the points.
Example: Graph in the -plane.
Try it!
How do I identify features of parabolas from quadratic functions?
Forms & features of quadratic functions
Standard form, factored form, and vertex form: What forms do quadratic equations take?
For all three forms of quadratic equations, the coefficient of the -term, , tells us whether the parabola opens upward or downward:
- If
, then the parabola opens upward. - If
, then the parabola opens downward.
The magnitude of also describes how steep or shallow the parabola is. Parabolas with larger magnitudes of are more steep and narrow compared to parabolas with smaller magnitudes of , which tend to be more shallow and wide.
The graph below shows the graphs of for various values of .
The standard form of a quadratic equation, , shows the -intercept of the parabola:
- The
-intercept of the parabola is located at .
The factored form of a quadratic equation, ,
shows the -intercept(s) of the parabola:
and are solutions to the equation .- The
-intercepts of the parabola are located at and . - The terms
-intercept, zero, and root can be used interchangeably.
The vertex form of a quadratic equation, , reveals the vertex of the parabola.
- The vertex of the parabola is located at
.
To identify the features of a parabola from a quadratic equation:
- Remember which equation form displays the relevant features as constants or coefficients.
- Rewrite the equation in a more helpful form if necessary.
- Identify the constants or coefficients that correspond to the features of interest.
Example: What are the zeros of the graph of ?
To match a parabola with its quadratic equation:
- Determine the features of the parabola.
- Identify the features shown in quadratic equation(s).
- Select a quadratic equation with the same features as the parabola.
- Plug in a point that is not a feature from Step 2 to calculate the coefficient of the
-term if necessary.
Example:
What is a possible equation for the parabola shown above?
Try it!
How do I rewrite quadratic functions to reveal specific features of parabolas?
Equivalent forms of quadratic functions
When we're given a quadratic function, we can rewrite the function according to the features we want to display:
-intercept: standard form -intercept(s): factored form- Vertex: vertex form
When rewriting a quadratic function to display specific graphical features:
- Choose the appropriate quadratic form based on the graphic feature to be displayed.
- Rewrite the given quadratic expression as an equivalent expression in the form identified in Step 1.
Example:
The graph of is shown above. Write an equivalent equation from which the coordinates of the vertex can be identified as constants in the equation.
Try it!
How do I transform graphs of quadratic functions?
Intro to parabola transformations
Translating, stretching, and reflecting: How does changing the function transform the parabola?
We can use function notation to represent the translation of a graph in the -plane. If the graph of is graphed in the -plane and is a positive constant:
- The graph of
is the graph of shifted to the right by units. - The graph of
is the graph of shifted to the left by units. - The graph of
is the graph of shifted up by units. - The graph of
is the graph of shifted down by units.
The graph below shows the graph of the quadratic function alongside various translations:
- The graph of
translates the graph of units to the right. - The graph of
translates the graph units to the left. - The graph of
translates the graph units up. - The graph of
translates the graph units down.
We can also represent stretching and reflecting graphs algebraically. If the graph of is graphed in the -plane and is a positive constant:
- The graph of
is the graph of reflected across the -axis. - The graph of
is the graph of reflected across the -axis. - The graph of
is the graph of stretched vertically by a factor of .
The graph below shows the graph of the quadratic function alongside various transformations:
- The graph of
is the graph of reflected across the -axis. - The graph of
is the graph of reflected across the -axis. - The graph of
is the graph of stretched vertically by a factor of .
Try it!
Your turn!
Things to remember
Forms of quadratic equations
Standard form: A parabola with the equation has its -intercept located at .
Factored form: A parabola with the equation has its -intercept(s) located at and .
Vertex form: A parabola with the equation has its vertex located at .
When we're given a quadratic function, we can rewrite the function according to the features we want to display:
-intercept: standard form -intercept(s): factored form- Vertex: vertex form
Transformations
If the graph of is graphed in the -plane and is a positive constant:
- The graph of
is the graph of shifted to the right by units. - The graph of
is the graph of shifted to the left by units. - The graph of
is the graph of shifted up by units. - The graph of
is the graph of shifted down by units. - The graph of
is the graph of reflected across the -axis. - The graph of
is the graph of reflected across the -axis. - The graph of
is the graph of stretched vertically by a factor of .
Want to join the conversation?
- when life gives you lemons(66 votes)
- wow! nothing made sense! yay! :D SAT is in one less than month AND I DONT UNDERSTAND PARABOLAS! T^T(74 votes)
- august 26th DSAT and i dont understand transformations(6 votes)
- guys i would recommend searching desmos calculator on google and opening it on other tab, because it is used for graphing and it is also used for digital SAT hence it is very useful to get a perfect score in your math section
all the best lads yall got this(56 votes)- Encouraging words of wisdom!(2 votes)
- For those, who face difficulty to understand this lesson - Please search for extra content on Youtube. This lesson needs lots of background knowledge. So, whichever term or topic you feel is tough search on Youtube and gain some basic knowledge about that. It takes me 2 days to complete this lesson! But eventually, I understand everything :) Try hard, never give up!(28 votes)
- Now I have to admit. It is the hardest chapter in entire Math section so far.(27 votes)
- tbh, along with this there are more such lessons(1 vote)
- this lesson actually has me crying. i hate quadratic functions so much :((23 votes)
- What is this transformation oh god forgive me 🤧(20 votes)
- i didn't understand ANYTHING!!(12 votes)
- Try the quadratics unit in Khan Maths.(5 votes)
- is this course really helpful to get a good sat score(10 votes)