Part A: Determine whether the parabola opens up or down. Part B: Write the equation of the axis of symmetry. Part C: Find the coordinates of the verte...
Name: __________________ Lesson 9-‐A Notes –Algebra 1 Graphing Quadratic Functions in Standard Form Date: __________________
TNSS F-‐IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. TNSS F-‐IF. 4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. TNSS F-‐IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. (Graph linear and quadratic functions and show intercepts, maxima, and minima.)
Past Target I graphed linear functions.
Present Target I can graph quadratic functions.
Future Target I will solve quadratic equations by graphing.
Standard Form of a quadratic function: 𝒇 𝒙 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄
1. Graph 𝑓 (𝑥) = −𝑥 . What changes when 𝒂 becomes negative?
2. Graph 𝑓 (𝑥) = 4𝑥 . What changed when 𝒂 > 𝟏?
3. Graph 𝑓 (𝑥)
x y -‐2 -‐1 0 1 2
Using a graphing calculator:
2
= 𝑥 . What changed when 𝟎 < 𝒂 < 𝟏? 3
This is called the “Parent graph” of quadratic functions.
• ___________________________ ________________________ are nonlinear and can be written in the form f (x) = ax 2 + bx + c , where a ≠ 0 . This form is called ___________________ ____________ of a quadratic function. • The shape of a quadratic function is called a _______________________________________. • Parabolas are ___________________ about a line called the ___________ _____ ___________________. • The maximum or minimum point of a parabola is called the ______________________. • If a is ________________, then the parabola opens upward and the vertex is the ___________________ point of the graph (lowest point). • If a is ________________, then the parabola opens downward and the vertex is the __________________ point of the graph (highest point).
Part A: Determine whether the parabola opens up or down. Part B: Write the equation of the axis of symmetry. Part C: Find the coordinates of the vertex. Ex. 𝒚 = 𝒙𝟐 − 𝟒𝒙 + 𝟑 . Part D: Identify the vertex as a maximum or minimum. Part E: Graph the function. Part F: State the domain and the range of the function. -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ Ex. 𝒚 = 𝒙𝟐 − 𝟒𝒙 + 𝟑 Part A: Visualize the parabola If the leading coefficient is POSITIVE – The parabola opens ___. (Think ________________) If the leading coefficient is NEGATIVE – The parabola opens _______. (Think _______________) Part B: To find the equation of the axis of symmetry , use 𝑥
=−
9
.:
𝑜𝑟 𝑥 =
>9 .:
.
The axis of symmetry is a vertical line. Part C: To find the vertex of the parabola, substitute the x-‐coordinate found in Part B into the equation and solve for y. Write the vertex as an ordered pair. Part D: Identify the vertex as a maximum or minimum. (Maximum means it’s at the TOP, Minimum means it’s at the BOTTOM) Part E: Graph the function. Step 1: Sketch the axis of symmetry. Step 2: Plot the vertex from Part C. Step 3: To find a 2nd point on the graph, substitute another number for x into the original equation. Step 4: Use reflection through the axis of symmetry to find the 3rd point. Step 5: Repeat steps 3 and 4 to find other points on the graph. Step 6: Connect the points with a smooth curve. Part F: State the domain and range of the function. Domain: X Values that are included in the graph Range: Y Values that are included in the graph
Practice with Graphing Quadratics 1. 𝑓 𝑥 = −𝑥 . + 10𝑥 − 13 a =
b = c =
a. Direction of opening is ______ Happy J or Sad L b. Axis of symmetry: _________ c. Vertex _________ d. The vertex is a ________point (max. or min.)
e. Make a table: f. Domain: Range:
2. 𝑓 𝑥 = 3𝑥 . − 12𝑥 + 5 a =
b = c =
a. Direction of opening is ______ Happy J or Sad L b. Axis of symmetry: _________
c. Vertex _________ d. The vertex is a ________point (max. or min.) e. Make a table: f. Domain: Range:
3. 𝑓 𝑥 = 𝑥 . + 3 a =
b = c =
a. Direction of opening is ______ Happy J or Sad L b. Axis of symmetry: _________ c. Vertex _________ d. The vertex is a ________point (max. or min.)
e. Make a table: f. Domain: Range:
4. 𝑓 𝑥 = −2𝑥 . − 8𝑥 − 3
b = c =
a. Direction of opening is ______ Happy J or Sad L b. Axis of symmetry: _________ c. Vertex _________ d. The vertex is a ________point (max. or min.)
a =
e. Make a table: f. Domain: Range:
5. 𝑓 𝑥 = 𝑥 . − 6𝑥 + 5 a =
b = c =
a. Direction of opening is ______ Happy J or Sad L b. Axis of symmetry: _________ c. Vertex _________ d. The vertex is a ________point (max. or min.)
e. Make a table: f. Domain: Range:
6.
𝑓 𝑥 = −𝑥 . + 2𝑥 + 3 a =
b = c =
a. Direction of opening is ______ Happy J or Sad L b. Axis of symmetry: _________ c. Vertex _________ d. The vertex is a ________point (max. or min.) e. Make a table: f. Domain: Range:
7. 𝑓 𝑥 = −4𝑥 . a =
b = c =
a. Direction of opening is ______ Happy J or Sad L b. Axis of symmetry: _________ c. Vertex _________ d. The vertex is a ________point (max. or min.)
e. Make a table: f. Domain: Range:
8. y
=
1 2 x + 2 x 2 a =
b = c =
a. Direction of opening is ______ Happy J or Sad L b. Axis of symmetry: _________ c. Vertex _________ d. The vertex is a ________point (max. or min.)
