Connecting Quadratics - Lesson 2.1

These flashcards cover key concepts related to quadratic functions and parabolas, including vertex form, symmetry, and equations.

For two parabolas, f(x) = 3(x - 1)^2 + 2 and g(x) = -4(x + 5)^2 - 1, describe how the sign of a determines the direction of opening for each?

For f(x), a = 3 (positive) means it opens upwards. For g(x), a = -4 (negative) means it opens downwards.

Compare the width of the parabolas f(x) = 0.5(x - 2)^2 + 3 and g(x) = 2(x - 2)^2 + 3. Which one appears wider and why?

The parabola f(x) is wider because its |a| value (0.5) is between 0 and 1. The parabola g(x) is narrower because its |a| value (2) is greater than 1.

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The general equation is f(x) = ax^2 + bx + c, where a \neq 0.

Given the quadratic function f(x) = 3x^2 - 12x + 5, use the appropriate formula to find the x-coordinate of its vertex.

The x-coordinate of the vertex is x = -b / (2a). For f(x), x = -(-12) / (2 * 3) = 12 / 6 = 2.

Without converting forms, what is the y-intercept of the parabola described by f(x) = -2x^2 + 7x - 4?

The y-intercept is the constant term c, which is -4. It is at (0, -4).

Identify the x-intercepts of the parabola represented by the equation f(x) = 2(x - 7)(x + 1)

The x-intercepts are r1 = 7 and r2 = -1.

Convert the vertex form f(x) = 2(x - 3)^2 + 1 into its equivalent standard form.

f(x) = 2(x^2 - 6x + 9) + 1 = 2x^2 - 12x + 18 + 1 = 2x^2 - 12x + 19

Transform the factored form f(x) = (x + 2)(x - 4) into its standard form.

f(x) = x^2 - 4x + 2x - 8 = x^2 - 2x - 8

Convert the standard form quadratic equation f(x) = x^2 - x - 6 into its factored form.

f(x) = (x - 3)(x + 2).

If a parabola has x-intercepts at x = -3 and x = 9, determine the equation of its axis of symmetry.

The axis of symmetry is halfway between the x-intercepts. So, x = (-3 + 9) / 2 = 3. The equation is x = 3.

For two parabolas, f(x) = 3(x - 1)^2 + 2 and g(x) = -4(x + 5)^2 - 1, describe how the sign of a determines the direction of opening for each?\n\n

For f(x), a = 3 (positive) means it opens upwards. For g(x), a = -4 (negative) means it opens downwards.\n\n

Compare the width of the parabolas f(x) = 0.5(x - 2)^2 + 3 and g(x) = 2(x - 2)^2 + 3. Which one appears wider and why?\n\n

The parabola f(x) is wider because its \|a\| value (0.5) is between 0 and 1. The parabola g(x) is narrower because its \|a\| value (2) is greater than 1.\n\n

A quadratic function has the highest exponent of 2. Write its general equation in standard form and state the condition for its leading coefficient.\n\n

The general equation is f(x) = ax^2 + bx + c, where a \neq 0.\n\n

Given the quadratic function f(x) = 3x^2 - 12x + 5, use the appropriate formula to find the x-coordinate of its vertex.\n\n

The x-coordinate of the vertex is x = -b / (2a). For f(x), x = -(-12) / (2 \* 3) = 12 / 6 = 2.\n\n

Without converting forms, what is the y-intercept of the parabola described by f(x) = -2x^2 + 7x - 4?\n\n

The y-intercept is the constant term c, which is -4. It is at (0, -4).\n\n

Identify the x-intercepts of the parabola represented by the equation f(x) = 2(x - 7)(x + 1)\n\n

The x-intercepts are r1 = 7 and r2 = -1.\n\n

Convert the vertex form f(x) = 2(x - 3)^2 + 1 into its equivalent standard form.\n\n

f(x) = 2(x^2 - 6x + 9) + 1 = 2x^2 - 12x + 18 + 1 = 2x^2 - 12x + 19\n\n

Transform the factored form f(x) = (x + 2)(x - 4) into its standard form.\n\n

f(x) = x^2 - 4x + 2x - 8 = x^2 - 2x - 8\n\n

Convert the standard form quadratic equation f(x) = x^2 - x - 6 into its factored form.\n\n

f(x) = (x - 3)(x + 2).\n\n

If a parabola has x-intercepts at x = -3 and x = 9, determine the equation of its axis of symmetry.\n\n

The axis of symmetry is halfway between the x-intercepts. So, x = (-3 + 9) / 2 = 3. The equation is x = 3.\n\n

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The vertex form is f(x) = a(x - h)^2 + k. (h, k) represents the coordinates of the vertex.\n\n

What is the general equation for a quadratic function in factored form, and what do r1 and r2 signify?\n\n

The factored form is f(x) = a(x - r1)(x - r2). r1 and r2 are the x-intercepts (roots) of the parabola.\n\n

How can you determine if a quadratic function has a maximum or minimum value, and where is this value located?\n\n

If a > 0, the parabola opens upwards and has a minimum value at the y-coordinate of the vertex. If a < 0, it opens downwards and has a maximum value at the y-coordinate of the vertex.\n\n

What are the domain and range for the quadratic function f(x) = a(x - h)^2 + k?\n\n

The domain is all real numbers, denoted as (-\infty, \infty). The range is [k, \infty) if a > 0, or (-\infty, k] if a < 0.\n\n