Lesson 3: Quadratic Functions
Estimated time: 40-50 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Define quadratic functions and identify their key features
- Recognize and work with standard form: y = ax² + bx + c
- Recognize and work with vertex form: y = a(x - h)² + k
- Identify the vertex, axis of symmetry, and direction of opening
- Determine whether a parabola opens upward or downward
- Find the vertex from standard form using the formula h = -b/(2a)
- Transform quadratic functions (shifts, reflections, stretches/compressions)
- Graph parabolas using vertex and key points
What is a Quadratic Function?
While linear functions create straight lines, quadratic functions create beautiful U-shaped curves called parabolas. These curves appear everywhere in nature and physics—from the path of a thrown ball to satellite dish shapes to suspension bridge cables.
Quadratic Function: A function that can be written in the form
f(x) = ax² + bx + c
where a, b, and c are constants and a ≠ 0.
The graph of a quadratic function is called a parabola.
Why must a ≠ 0?
If a = 0, the x² term disappears, and you're left with bx + c, which is a linear function (a line, not a parabola). The x² term is what makes it quadratic!
Key Features of Parabolas
Vertex
The highest or lowest point on the parabola (the "turning point")
Coordinates: (h, k)
Axis of Symmetry
A vertical line through the vertex that divides the parabola into mirror images
Equation: x = h
Direction
Parabola opens upward or downward
Determined by sign of a
Y-Intercept
Where the parabola crosses the y-axis
Coordinates: (0, c)
Standard Form of Quadratic Functions
Standard Form
where:
a determines direction and width of parabola
b affects the position of the vertex
c is the y-intercept
The Coefficient "a": Direction and Width
a > 0 (positive)
Parabola opens upward (U-shape)
Vertex is a minimum point
Example: y = 2x²
a < 0 (negative)
Parabola opens downward (∩-shape)
Vertex is a maximum point
Example: y = -3x²
|a| > 1
Parabola is narrower (steeper)
Vertical stretch
Example: y = 4x² is narrower than y = x²
0 < |a| < 1
Parabola is wider (flatter)
Vertical compression
Example: y = 0.5x² is wider than y = x²
Example 1: Identifying Features from Standard Form
For f(x) = 2x² - 8x + 3, identify:
(a) Does it open upward or downward?
(b) What is the y-intercept?
(c) Is it wider or narrower than y = x²?
Solution:
Comparing to y = ax² + bx + c: a = 2, b = -8, c = 3
(a) Since a = 2 > 0, the parabola opens upward. The vertex is a minimum.
(b) The y-intercept is c = 3, at point (0, 3).
(c) Since |a| = 2 > 1, the parabola is narrower than y = x².
Finding the Vertex from Standard Form
When a quadratic is in standard form y = ax² + bx + c, we can find the vertex using a formula:
Vertex Formula
For y = ax² + bx + c, the vertex is at (h, k) where:
Then substitute h into the equation to find k:
Vertex: (h, k) = (-b/(2a), f(-b/(2a)))
Axis of symmetry: x = -b/(2a)
Example 2: Finding the Vertex
Find the vertex of f(x) = x² - 6x + 5.
Solution:
Identify: a = 1, b = -6, c = 5
Step 1: Find h using h = -b/(2a)
h = -(-6)/(2·1)
h = 6/2
h = 3
Step 2: Find k by substituting h = 3 into the function
k = f(3) = (3)² - 6(3) + 5
k = 9 - 18 + 5
k = -4
Answer: Vertex is at (3, -4)
Axis of symmetry: x = 3
Since a = 1 > 0, the parabola opens upward, so (3, -4) is a minimum point.
Example 3: Complete Analysis
Analyze f(x) = -2x² + 8x - 3:
(a) Direction of opening
(b) Vertex
(c) Axis of symmetry
(d) Y-intercept
(e) Maximum or minimum value
Solution:
a = -2, b = 8, c = -3
(a) Since a = -2 < 0, opens downward
(b) Finding vertex:
h = -b/(2a) = -8/(2·(-2)) = -8/(-4) = 2
k = f(2) = -2(2)² + 8(2) - 3 = -8 + 16 - 3 = 5
Vertex: (2, 5)
(c) Axis of symmetry: x = 2
(d) Y-intercept: (0, -3)
(e) Since parabola opens downward, the vertex is a maximum. The maximum value is y = 5 at x = 2.
Vertex Form of Quadratic Functions
The vertex form immediately shows you the vertex and transformations applied to the parent function y = x².
Vertex Form
where:
(h, k) is the vertex
a determines direction and width (same as in standard form)
Axis of symmetry: x = h
Important: Watch the Signs!
In y = a(x - h)² + k:
- If you see (x - 3)², then h = +3 (vertex moves RIGHT)
- If you see (x + 3)², then h = -3 (vertex moves LEFT)
- If you see ...+ 5, then k = +5 (vertex moves UP)
- If you see ...- 5, then k = -5 (vertex moves DOWN)
Example 4: Identifying Vertex from Vertex Form
Identify the vertex for each function:
(a) y = (x - 4)² + 7
a = 1, h = 4, k = 7
Vertex: (4, 7)
Opens upward (a > 0)
(b) y = -3(x + 2)² - 5
a = -3, h = -2, k = -5
Vertex: (-2, -5)
Opens downward (a < 0)
(c) y = 0.5(x - 1)² + 3
a = 0.5, h = 1, k = 3
Vertex: (1, 3)
Opens upward, wider than y = x²
Example 5: Converting Standard Form to Vertex Form
We found earlier that y = x² - 6x + 5 has vertex (3, -4). Write this in vertex form.
Solution:
Vertex is (h, k) = (3, -4), and a = 1
Substitute into y = a(x - h)² + k:
y = 1(x - 3)² + (-4)
y = (x - 3)² - 4
Verify: Expand to check
y = (x - 3)² - 4
y = x² - 6x + 9 - 4
y = x² - 6x + 5
Transformations of Quadratic Functions
Understanding transformations helps you graph quadratics quickly by starting with the parent function y = x² and applying shifts, reflections, and stretches.
Transformation Guide: y = a(x - h)² + k
Vertical Shift
+k: Shift UP k units
-k: Shift DOWN k units
Example: y = x² + 3 shifts up 3
Horizontal Shift
(x - h): Shift RIGHT h units
(x + h): Shift LEFT h units
Example: y = (x - 2)² shifts right 2
Vertical Stretch/Compression
|a| > 1: Stretch (narrower)
0 < |a| < 1: Compress (wider)
Example: y = 3x² is narrower
Reflection
a > 0: Opens upward
a < 0: Opens downward (reflected over x-axis)
Example: y = -x² flips upside down
Example 6: Describing Transformations
Describe how to obtain y = -2(x + 3)² + 5 from the parent function y = x².
Solution:
Starting with y = x², apply transformations in this order:
1. Horizontal shift LEFT 3 units (x + 3)
y = (x + 3)²
2. Vertical stretch by factor of 2 (coefficient 2)
y = 2(x + 3)²
3. Reflection over x-axis (negative sign)
y = -2(x + 3)²
4. Vertical shift UP 5 units (+5)
y = -2(x + 3)² + 5
Summary: Shift left 3, stretch vertically by 2, reflect over x-axis, shift up 5
Vertex: (-3, 5), opens downward
Example 7: Writing Function from Transformations
Write the equation of the parabola obtained by:
- Shifting y = x² right 4 units
- Shifting down 6 units
- Reflecting over the x-axis
Solution:
Right 4: h = 4, so (x - 4)
Down 6: k = -6
Reflection: a = -1
Vertex form: y = a(x - h)² + k
y = -(x - 4)² - 6
Or expanded to standard form:
y = -(x² - 8x + 16) - 6
y = -x² + 8x - 16 - 6
y = -x² + 8x - 22
Graphing Parabolas
Steps to Graph a Parabola
- Find the vertex (use formula or read from vertex form)
- Determine direction (upward if a > 0, downward if a < 0)
- Find the axis of symmetry (vertical line x = h)
- Find the y-intercept (set x = 0)
- Find additional points (use symmetry to mirror points)
- Sketch the parabola through all points
Example 8: Graphing from Vertex Form
Graph y = (x - 2)² - 1.
Solution:
Step 1: Vertex is (h, k) = (2, -1)
Step 2: Since a = 1 > 0, opens upward
Step 3: Axis of symmetry: x = 2
Step 4: Y-intercept (x = 0):
y = (0 - 2)² - 1 = 4 - 1 = 3
Point: (0, 3)
Step 5: Additional points using symmetry:
When x = 1: y = (1-2)² - 1 = 1 - 1 = 0, point (1, 0)
By symmetry, (3, 0) is also on the graph
By symmetry with (0, 3), point (4, 3) is on the graph
Key Points to Plot:
Vertex: (2, -1)
Y-intercept: (0, 3)
X-intercepts: (1, 0) and (3, 0)
Additional: (4, 3)
Draw a smooth U-shaped curve through these points.
Example 9: Graphing from Standard Form
Graph f(x) = -x² + 4x - 3.
Solution:
a = -1, b = 4, c = -3
Step 1: Find vertex:
h = -b/(2a) = -4/(2·(-1)) = -4/(-2) = 2
k = f(2) = -(2)² + 4(2) - 3 = -4 + 8 - 3 = 1
Vertex: (2, 1)
Step 2: Opens downward (a = -1 < 0)
Step 3: Axis of symmetry: x = 2
Step 4: Y-intercept: (0, -3)
Step 5: Additional point (x = 1):
f(1) = -(1)² + 4(1) - 3 = -1 + 4 - 3 = 0
Point: (1, 0)
By symmetry: (3, 0)
Key Points: Vertex (2, 1), Y-intercept (0, -3), X-intercepts (1, 0) and (3, 0), and by symmetry (4, -3)
Draw a smooth ∩-shaped curve opening downward.
Check Your Understanding
Try these questions to see if you've grasped the key concepts:
1. For f(x) = -3x² + 12x - 5, does the parabola open upward or downward? Is the vertex a maximum or minimum?
a = -3 < 0, so the parabola opens downward.
Since it opens downward, the vertex is a maximum point.
2. Find the vertex of f(x) = x² + 8x + 7.
a = 1, b = 8, c = 7
h = -b/(2a) = -8/(2·1) = -8/2 = -4
k = f(-4) = (-4)² + 8(-4) + 7
k = 16 - 32 + 7
k = -9
Vertex: (-4, -9)
Axis of symmetry: x = -4
3. What is the vertex of y = 2(x + 5)² - 3?
This is in vertex form: y = a(x - h)² + k
y = 2(x - (-5))² + (-3)
h = -5, k = -3
Vertex: (-5, -3)
The parabola opens upward (a = 2 > 0) and is narrower than y = x².
4. Describe the transformations applied to y = x² to get y = -0.5(x - 3)² + 4.
1. Shift right 3 units (x - 3)
2. Vertical compression by factor 0.5 (makes it wider)
3. Reflection over x-axis (negative sign, opens downward)
4. Shift up 4 units (+4)
Vertex: (3, 4), opens downward
5. Write the equation in vertex form for a parabola with vertex (1, -6) that opens upward and has the same width as y = x².
Vertex form: y = a(x - h)² + k
h = 1, k = -6
Same width as y = x² means a = 1
Opens upward confirms a = 1 (positive)
Answer: y = (x - 1)² - 6
6. For y = x² - 4x + 3, find the vertex and y-intercept.
h = -b/(2a) = -(-4)/(2·1) = 4/2 = 2
k = (2)² - 4(2) + 3 = 4 - 8 + 3 = -1
Vertex: (2, -1)
Y-intercept: c = 3, so (0, 3)
7. What is the axis of symmetry for f(x) = 2x² - 12x + 7?
The axis of symmetry passes through the vertex.
h = -b/(2a) = -(-12)/(2·2) = 12/4 = 3
Axis of symmetry: x = 3
Key Takeaways
- Quadratic function: f(x) = ax² + bx + c where a ≠ 0
- Graph of a quadratic is a parabola (U-shaped curve)
- a > 0: opens upward (minimum); a < 0: opens downward (maximum)
- |a| > 1: narrower; 0 < |a| < 1: wider
- Standard form: y = ax² + bx + c (y-intercept is c)
- Vertex form: y = a(x - h)² + k (vertex is (h, k))
- Vertex from standard form: h = -b/(2a), then k = f(h)
- Axis of symmetry: vertical line x = h through vertex
- Transformations: horizontal/vertical shifts, reflections, stretches/compressions
Ready for More?
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In Lesson 4, you'll learn how to solve quadratic equations using factoring, the quadratic formula, and completing the square. You'll also learn about the discriminant and its role in determining the number of solutions!
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