Unit 14: Quadratic Functions and Graphs
Every quadratic graphs as a parabola -- its vertex, its width, and which way it opens all come straight from the coefficients.
6 topics · 20 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
The Shape of a Parabola
- How it worksHow does squaring x make a parabola's two sides mirror each other perfectly?
- Worked exampleGraph y = x2 - 4 by plotting points on both sides of the axis of symmetry.
- Common mistakeWhy doesn't a parabola ever cross its own axis of symmetry?
Your turn1 of 2
For y = x2 - 7, find y when x = 5.The Vertex and Axis of Symmetry
- How it worksHow does the formula -b over 2a locate a parabola's axis of symmetry straight from standard form?
- Worked exampleFind the vertex of y = x2 - 6x + 5.
- Common mistakeWhy doesn't the vertex of a parabola always sit at the y-intercept?
Your turn1 of 2
Find the x-coordinate of the vertex of y = 2x2 + 10x + 19.Transformations of Quadratic Functions
- How it worksHow do the numbers in y = a times the quantity x minus h, squared, plus k, each move or reshape the basic parabola y = x2?
- Worked exampleShow me how the graph of y = -2 times the quantity x minus 3, squared, plus 1, is transformed from y = x2.
- Going furtherWhat happens to a parabola's width as the leading coefficient a grows from 1 toward 10?
- Going furtherWhat happens to a parabola's opening direction when the leading coefficient a switches from positive to negative?
Your turn1 of 2
A parabola y = x2 is scaled by a factor of -2, shifted right by 6, and up by 8. Write the resulting function y(x).Graphing from Vertex Form
- How it worksHow do you read the vertex and stretch directly out of an equation written in vertex form?
- Worked exampleGraph y = 2 times the quantity x plus 1, squared, minus 3, using its vertex form.
- Common mistakeWhy is h subtracted in vertex form even when the vertex's x-coordinate is negative?
Your turn1 of 2
For y = 4 times the quantity x plus 6, squared, minus 8, find the vertex's y-coordinate.Graphing from Standard Form
- How it worksHow do the vertex formula and the y-intercept together let you sketch a parabola straight from standard form?
- Worked exampleGraph y = x2 + 2x - 3 by finding its vertex, intercepts, and axis of symmetry.
- Ties togetherHow does graphing from standard form connect back to completing the square to reach vertex form?
Your turn1 of 2
For y = x2 - (-4+7)x + -4·7, find the smaller x-intercept.Comparing Quadratic Functions
- How it worksHow do you find a parabola's maximum or minimum value from a table of its outputs?
- Worked exampleCompare the vertex of y = x2 - 4x + 1 to the vertex of a parabola whose table has a minimum at (3, -2).
- Going furtherWhat happens when you compare two parabolas that have the same vertex but different leading coefficients?
- Going furtherWhat happens to a parabola's vertex when its standard-form coefficient b changes sign?
Your turn1 of 2
The parabola y = x2 + 8x + 15 has its vertex compared to a second parabola whose table shows a minimum at (8, -4). Find the first parabola's vertex y-coordinate minus -4.