Pre-Cal IXL C.1-9 Test Flashcards

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Questions and Answers

The maximum value of a parabola that opens downwards is the y coordinate of the ____ point on the graph.

highest

The minimum value of a parabola that opens upwards is the y coordinate of the ____ point on the graph.

lowest

What is the x coordinate of the vertex of the equation y = ax^2 + bx + c?

x = -b/2a

How do you find the y coordinate of the vertex of the equation y = ax^2 + bx + c using the value of x?

<p>Plug x = -b/2a into the equation.</p> Signup and view all the answers

Find the minimum value of the parabola y = x^2 + 6.

<p>6</p> Signup and view all the answers

Find the minimum value of the parabola y = 3x^2 - 6x.

<p>-3</p> Signup and view all the answers

The vertex of a parabola that opens upwards is the ____ point.

<p>lowest</p> Signup and view all the answers

The vertex of a parabola that opens downwards is the ____ point.

<p>highest</p> Signup and view all the answers

What is the y coordinate of the point where the parabola intersects the y axis?

<p>y intercept</p> Signup and view all the answers

What is the line that passes through the vertex and divides the parabola into two halves that are mirror images?

<p>axis of symmetry</p> Signup and view all the answers

Find the y-intercept of the parabola y = x^2 + 19/5.

<p>(0, 19/5)</p> Signup and view all the answers

Find the equation of the axis of symmetry for the parabola y = x^2 - 4x.

<p>x = 2</p> Signup and view all the answers

What is the graph of a quadratic polynomial called?

<p>parabola</p> Signup and view all the answers

What is the vertex form of a parabola that opens up or down?

<p>y = a(x-h)^2 + k</p> Signup and view all the answers

When the equation is written in vertex form, the vertex is (h, ____).

<p>k</p> Signup and view all the answers

(In vertex form) If a > 0, the parabola opens ____.

<p>up</p> Signup and view all the answers

(In vertex form) If a < 0, the parabola opens ____.

<p>down</p> Signup and view all the answers

Match the graph to its function:

<p>a) -x^2 - 5x - 6 = =(x + 3)(x + 2) b) h(x) = 2x^2 - 24x + 70 = =(x - 5)(x - 7)</p> Signup and view all the answers

Solve by using square roots: 181z^2 - 134 = -157z^2 - 132. What are the solutions?

<p>z = +/- 1/13</p> Signup and view all the answers

Solve by factoring: 4y^2 - 36y + 81 = 0. What is the solution?

<p>y = 9/2</p> Signup and view all the answers

Solve by completing the square: m^2 - 30m - 1 = 0. What are the solutions?

<p>m = 30.03, -0.03</p> Signup and view all the answers

Solve using the quadratic formula: 2j^2 = -3j + 3. What are the solutions?

<p>varies</p> Signup and view all the answers

Find the discriminant for the equation 4u = -7u^2 - 6. What type of solutions does this equation have?

<p>(-152) two complex (non-real) solutions</p> Signup and view all the answers

What are the steps for solving by completing the square?

<ol> <li>Make the left side = x^2 + bx 2. Add (b/2)^2 to both sides 3. Factor the left side as (x + b/2)^2 4. Take the square roots of both sides and solve.</li> </ol> Signup and view all the answers

What are the steps for solving by factoring?

<ol> <li>Write in standard form 2. Factor 3. Zero product property rule (set equal to 0)</li> </ol> Signup and view all the answers

What is this? (the quadratic formula)

<p>Used to find the roots of a quadratic equation.</p> Signup and view all the answers

The term underneath the square in the quadratic formula is known as the ____.

<p>discriminant</p> Signup and view all the answers

The discriminant tells the number of ____ of solutions to a quadratic.

<p>solutions</p> Signup and view all the answers

What is the discriminant formula?

<p>b^2 - 4ac</p> Signup and view all the answers

If the solution to the discriminant is > 0, then there are... ____ roots.

<p>2 real</p> Signup and view all the answers

If the solution to the discriminant is = 0, then there is... ____ root.

<p>1 real</p> Signup and view all the answers

If the solution to the discriminant is < 0, then there are... ____ roots.

<p>no real (two complex/imaginary)</p> Signup and view all the answers

Since there are two real roots, the discriminant is... ____.

<p>greater than 0</p> Signup and view all the answers

Since there is only one real root, the discriminant is... ____.

<p>equal to 0</p> Signup and view all the answers

Since there are no real roots, the discriminant is... ____.

<p>less than 0</p> Signup and view all the answers

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Study Notes

Parabola Characteristics

  • The highest point on a downward-opening parabola is the vertex's y-coordinate, known as the maximum value.
  • The lowest point on an upward-opening parabola is the vertex's y-coordinate, known as the minimum value.
  • The vertex of a parabola can be identified using the formula x = -b/2a, which represents the x-coordinate in the quadratic equation y = ax^2 + bx + c.

Finding Coordinates and Values

  • To find the y-coordinate of the vertex, substitute x = -b/2a into the original quadratic equation.
  • The minimum value of the parabola y = x^2 + 6 is 6, occurring at the vertex.
  • The minimum value of y = 3x^2 - 6x is -3, identified similarly.

Key Parabola Terminology

  • The vertex of an upward-opening parabola is termed the "lowest point."
  • The vertex of a downward-opening parabola is termed the "highest point."
  • The y-intercept is the point where the parabola crosses the y-axis, expressed as the y-coordinate at x = 0.
  • The axis of symmetry is a vertical line that runs through the vertex and divides the parabola into two symmetrical halves.

Vertex and Standard Forms

  • The standard form of a parabola can be written as y = a(x-h)^2 + k, where (h, k) is the vertex.
  • In vertex form, if a > 0, the parabola opens upwards; if a < 0, it opens downwards.

Solving Quadratic Equations

  • The quadratic formula, used for solving equations, is derived from standard form, and is essential when other methods are inapplicable.
  • The discriminant (b^2 - 4ac) determines the nature of the roots of quadratic equations:
    • Greater than 0: two distinct real roots.
    • Equal to 0: one real root.
    • Less than 0: two complex (non-real) roots.

Methods for Solving Quadratics

  • Completing the Square:
    • Steps include rewriting in standard form, factoring, and utilizing the zero product property.
  • Factoring:
    • Involves writing the equation in standard form, factoring into binomials, and applying the zero product property.

Solutions and Discriminants

  • The discriminant is crucial in understanding the quantity of solutions to a quadratic equation:
    • 0 indicates two real roots,

    • = 0 indicates one real root,
    • < 0 indicates no real roots (complex solutions).

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