Pre-Cal IXL C.1-9 Test Flashcards

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?

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

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

6 Signup and view all the answers

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

-3 Signup and view all the answers

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

lowest Signup and view all the answers

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

highest Signup and view all the answers

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

y intercept 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?

axis of symmetry Signup and view all the answers

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

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

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

x = 2 Signup and view all the answers

What is the graph of a quadratic polynomial called?

parabola Signup and view all the answers

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

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

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

k Signup and view all the answers

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

up Signup and view all the answers

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

down Signup and view all the answers

Match the graph to its function:

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

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

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

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

y = 9/2 Signup and view all the answers

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

m = 30.03, -0.03 Signup and view all the answers

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

varies Signup and view all the answers

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

(-152) two complex (non-real) solutions 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)

Used to find the roots of a quadratic equation. Signup and view all the answers

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

discriminant Signup and view all the answers

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

solutions Signup and view all the answers

What is the discriminant formula?

b^2 - 4ac Signup and view all the answers

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

2 real Signup and view all the answers

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

1 real Signup and view all the answers

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

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

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

greater than 0 Signup and view all the answers

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

equal to 0 Signup and view all the answers

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

less than 0 Signup and view all the answers

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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Description

Test your knowledge of key concepts in Pre-Calculation with these flashcards covering the highest and lowest points of parabolas, as well as the vertex formula. Perfect for students preparing for assessments in this fundamental area of mathematics.