Graphing Quadratic Functions Using Key Properties
Graphing quadratic functions using key properties is a Grade 11 Algebra 1 skill from enVision Chapter 8 following a 6-step process from standard form f(x) = ax^2 + bx + c. Step 1: determine opening direction from sign of a. Step 2: find axis of symmetry x = -b/(2a). Step 3: find vertex by substituting into the function. Step 4: find y-intercept at (0, c). Step 5: find the point symmetric to the y-intercept. Step 6: plot and sketch. For f(x) = x^2 + 2x - 8: opens up (a=1), axis x = -1, vertex (-1, -9), y-intercept (0,-8), symmetric point (-2,-8).
Key Concepts
To graph a quadratic function in standard form $f(x) = ax^2 + bx + c$: Step 1. Determine whether the parabola opens upward or downward using the sign of $a$. Step 2. Find the equation of the axis of symmetry: $x = \frac{b}{2a}$. Step 3. Find the vertex by substituting the axis of symmetry into the function. Step 4. Find the $y$ intercept at $(0, c)$. Step 5. Find the point symmetric to the $y$ intercept across the axis of symmetry. Step 6. Plot additional points as needed and sketch the parabola.
Common Questions
What are the key steps for graphing a quadratic function in standard form?
Find the opening direction (sign of a), axis of symmetry x = -b/2a, vertex, y-intercept at (0,c), its symmetric point, then sketch the parabola.
How do you find the axis of symmetry of f(x) = ax^2 + bx + c?
Use the formula x = -b/(2a). For f(x) = x^2 + 2x - 8, x = -2/(2*1) = -1.
How do you find the vertex of f(x) = x^2 + 2x - 8?
Axis of symmetry is x = -1. Substitute: f(-1) = 1 - 2 - 8 = -9. Vertex is (-1, -9).
What is the y-intercept of f(x) = -x^2 + 6x - 9?
c = -9, so y-intercept is (0, -9).
How do you find the point symmetric to the y-intercept?
Use the axis of symmetry to reflect (0, c). For f(x) = x^2 + 2x - 8 with axis x = -1, the symmetric point to (0, -8) is (-2, -8).
How do you determine if the parabola opens up or down?
If a > 0, the parabola opens upward and has a minimum vertex. If a < 0, it opens downward and has a maximum vertex.