![Picture](/uploads/2/6/2/1/26215381/116814207.jpg?250)
Curriculum Outcomes
Relations and Functions: Develop algebraic and graphical reasoning through the study of relations.
3. Analyze quadratic functions of the form ( )2 y = a x − p + q and determine the:
• vertex
• domain and range
• direction of opening
• axis of symmetry
• x- and y-intercepts.
[CN, R, T, V] [ICT: C6–4.3, C7–4.2]
4. Analyze quadratic functions of the form y = ax2 + bx + c to identifycharacteristics of the corresponding graph, including:
• vertex
• domain and range
• direction of opening
• axis of symmetry
• x- and y-intercepts
and to solve problems.
[CN, PS, R, T, V] [ICT: C6–4.1, C6–4.3]
5. Solve problems that involve quadratic equations.
[C, CN, PS, R, T, V] [ICT: C6–4.1]
Relations and Functions: Develop algebraic and graphical reasoning through the study of relations.
3. Analyze quadratic functions of the form ( )2 y = a x − p + q and determine the:
• vertex
• domain and range
• direction of opening
• axis of symmetry
• x- and y-intercepts.
[CN, R, T, V] [ICT: C6–4.3, C7–4.2]
4. Analyze quadratic functions of the form y = ax2 + bx + c to identifycharacteristics of the corresponding graph, including:
• vertex
• domain and range
• direction of opening
• axis of symmetry
• x- and y-intercepts
and to solve problems.
[CN, PS, R, T, V] [ICT: C6–4.1, C6–4.3]
5. Solve problems that involve quadratic equations.
[C, CN, PS, R, T, V] [ICT: C6–4.1]
![Picture](/uploads/2/6/2/1/26215381/published/quadratic-formula.png?250)
Things You Need to Know
- Quadratics are just polynomials that have a degree (highest exponent) of 2.
- The graph of quadratic will be either a right-side up U or an upside-down U.
- The places along the graph of a quadratic that cross the x-axis are called the x-intercepts, or more commonly, the roots.
- Vertex form:
- y = a(x - p)² + q
- The point (p, q) is the vertex of this quadratic. The vertex is the lowest or highest point on the graph (bottom of the "U").
- y = a(x - p)² + q
- Standard form:
- y = ax² + bx + c
- The value of c is the value of the y-intercept.
- The vertex isn't directly visible in this form. You can find the vertex using the formula x = -b/2a, and then plugging x into the standard form quadratic to find the y-coordinate of the vertex.
- y = ax² + bx + c
- You can change to standard form from vertex form by multiplying out the exponent.
- You can charge to vertex form by standard form by completing the square, OR by finding the vertex using the formula x = -b/2a, and then plugging that x into your original quadratic to find the y-coordinate of the vertex.
- The "a" value of both standard form and vertex form will be the same, regardless of what form it's in!
- The "a" value of both standard form and vertex form will be the same, regardless of what form it's in!
- There are four main ways to find the roots of a quadratic. Regardless of the method, make sure your quadratic is set to equal 0:
- Graphing calculator
- Factoring
- Vertex form, then rearranging to get x alone.
- Quadratic formula
- Not all quadratics will have roots. Some have 2 roots, some have 1 root, others have no roots at all. You can find the number of roots of a quadratic by finding the discriminant:
- b² - 4ac
- If this is greater than 0, there are 2 solutions.
- If this is equal to 0, there is 1 solution.
- If this is less than 0, there are no solutions.
- b² - 4ac
Video Tutorials
- Vertex form
- Standard form
- Completing the Square (going from standard to vertex form)
- Finding the Roots by Graphing
- Finding the Roots by Factoring
- How to Factor Any Quadratic
- In class, we called factoring a (ax² + bx + c) quadratic the "Adapted Sum/Product Rule". This video calls it "Decomposition". It's the same procedure either way.
- Finding the Roots using the Quadratic Formula