Quadratic Functions and Factoring Flashcards (39 cards)

In standard form, what determines a parabola’s opening direction and shape?

The leading coefficient $a$.

In standard form, what does $c$ give?

The y-intercept.

In vertex form, what are the coordinates of the vertex?

$(h,k)$.

In vertex form, what does $a$ indicate about the parabola?

Its opening direction and shape.

In factored form, what are the values that make the factors zero called?

The zeros or x-intercepts of the quadratic.

How do you begin expanding a squared binomial?

Write it as two identical factors.

How do you multiply the factors of a squared binomial?

Use FOIL.

What should you do after multiplying a squared binomial?

Simplify using the order of operations.

How is the greatest common factor of polynomial terms determined?

Use the greatest common coefficient and the lowest shared exponent of each variable.

How are the constants chosen when factoring a trinomial with leading coefficient 1?

Choose a factor pair of the constant term whose sum or difference gives the middle coefficient.

What product guides the split of the middle term in factoring $ax^2+bx+c$ when $a\ne1$?

The product $ac$.

What must the split coefficients of the middle term in $ax^2+bx+c$ satisfy when $a\ne1$?

They multiply to $ac$ and add to $b$.

After splitting the middle term in $ax^2+bx+c$, how are the four terms factored?

Group the terms and factor each group to obtain a common binomial.

What is done with the common binomial after grouping in factoring $ax^2+bx+c$?

Factor the common binomial once.

What terms define a difference of two squares?

Two perfect-square terms separated by subtraction.

How does a difference of two squares factor using its square roots?

As their difference times their sum.

What is the standard factorization formula for a difference of squares?

$A^2-B^2=(A+B)(A-B)$

What must the first and last terms of a perfect-square trinomial be?

Positive perfect squares.

How is the middle term of a perfect-square trinomial determined?

It equals twice the product of the first and last terms’ square roots.

How does a perfect-square trinomial factor, including its sign?

As a binomial squared, with the middle term’s sign.

What is the first step when solving a quadratic by factoring?

Put all terms on one side so the equation equals zero.

What comes after setting each factor equal to zero when solving by factoring?

Solve for the variable.

What determines whether a parabola opens upward or downward?

The sign of its leading coefficient.

Where does a quadratic graph change between increasing and decreasing?

At its vertex.

What do positive and negative intervals identify for a quadratic?

The x-values where its y-values are above or below the x-axis.

How are interval inequalities for positive and negative intervals written?

Strictly.

What is the axis of symmetry of $y=ax^2+bx+c$?

$x=-\frac{b}{2a}$.

How do you expand brackets when converting factored form to standard form?

Use FOIL.

What should you do with an outside coefficient when converting factored form to standard form?

Distribute it.

What should you do after expanding and distributing in factored form?

Collect like terms.

How does $9x^2-12x+4$ factor?

$(3x-2)^2$

What are the roots of $x^2-5x-50=0$?

$x=-10$ and $x=5$.

What factorization results from factoring $x^2-5x-50=0$?

$(x+10)(x-5)=0$.

What are the zeros of $h(t)=-5t^2+10t+70$?

$t=4$ and $t=-2$.

What factorization results from setting $h(t)=-5t^2+10t+70$ equal to zero?

$-5(t-4)(t+2)=0$.

What does an increasing interval mean for a quadratic graph?

Increasing x-values produce increasing y-values.

What does a decreasing interval mean for a quadratic graph?

Increasing x-values produce decreasing y-values.

What is the sign of second differences for an upward-opening parabola?

Positive.

What is the sign of second differences for a downward-opening parabola?

Negative.