Graphing Quadratic Functions in Intercept Form (Factored Form) - Algebra-2

1. Fundamental Concepts

Intercept/Factored Form of a Quadratic Function: The form is (where ), where p and q are the x-intercepts (roots) of the quadratic function, meaning the parabola crosses the x-axis at points and .
Parabola: The graph of a quadratic function in intercept/factored form is a U-shaped curve, similar to those in other forms.
Vertex: The minimum or maximum point of the parabola, lying on the axis of symmetry.
Axis of Symmetry: A vertical line that passes through the midpoint of the x-intercepts, dividing the parabola into two symmetrical halves.

Role of a:
- Determines the direction the parabola opens: If, the parabola opens upward (has a minimum vertex); if , it opens downward (has a maximum vertex).
- Affects the width of the parabola: Larger makes the parabola narrower; smaller makes it wider.
X-intercepts: Given directly by p and q in the intercept/factored form, so the parabola intersects the x-axis at and .
Axis of Symmetry: Calculated as the midpoint of the x-intercepts, with the formula . This line is crucial as the vertex lies on it.
Vertex: The x-coordinate of the vertex is the same as the axis of symmetry (); the y-coordinate is found by substituting this x-value back into the intercept/factored form function, i.e., .

Step 1: Identify x-intercepts: From the form , we have and , so the x-intercepts are and .
Step 2: Find the axis of symmetry: .
Step 3: Calculate the vertex: Substitute into the function: , so the vertex is .
Step 4: Determine the direction of opening: Since , the parabola opens downward.
Step 5: Plot points: Plot the x-intercepts , and the vertex , then connect them to form the parabola.
Key Features: X-intercepts , ; vertex ; axis of symmetry ; domain ; range ; maximum value 4.

Step 1: Identify x-intercepts: , , so x-intercepts are and .
Step 2: Find the axis of symmetry: .
Step 3: Calculate the vertex: Substitute into the function: , so the vertex is .
Step 4: Determine the direction of opening: Since, the parabola opens upward.
Step 5: Plot points: Plot the x-intercepts , and the vertex . To get more points, choose : , so the point is ; its symmetric point about is . Plot these and connect to form the parabola.
Key Features: X-intercepts , ; vertex ; axis of symmetry ; domain ; range ; minimum value -4.

Step 1: Identify x-intercepts: , , so x-intercepts are and .
Step 2: Find the axis of symmetry: .
Step 3: Calculate the vertex: Substitute into the function: , so the vertex is .
Step 4: Determine the direction of opening: Since , the parabola opens upward and is narrower than the parent function.
Step 5: Plot points: Plot the x-intercepts , and the vertex . Choose : , so the point is ; its symmetric point about is . Plot these and connect to form the parabola.
Key Features: X-intercepts , ; vertex ; axis of symmetry ; domain ; range ; minimum value -4.5.

Identify X-intercepts: From the intercept/factored form , directly obtain the x-intercepts and .
Calculate Axis of Symmetry: Use the formula to find the vertical line of symmetry.
Determine the Vertex: Substitute the x-coordinate of the axis of symmetry into the function to find the y-coordinate, getting the vertex .
Analyze the Direction and Width: Based on the value of a, determine if the parabola opens upward or downward and its width relative to the parent function.
Plot Additional Points: Choose a value for x (not an intercept) and calculate the corresponding y-value to get an extra point. Use the axis of symmetry to find its symmetric counterpart.
Sketch the Parabola: Plot the x-intercepts, vertex, and additional points, then draw a smooth curve through them to form the parabola.
Confirm Key Features: After graphing, verify the x-intercepts, vertex, axis of symmetry, domain, range, and minimum/maximum value.