07/14 7. Quadratic Functions and Equations Free Online FlashCards

What is the standard form of a quadratic function?

A quadratic function has the form $f(x)=ax^2+bx+c$, where $a\ne0$. Its graph is a parabola.

How does the sign of $a$ affect a parabola?

If $a>0$, the parabola opens upward and has a minimum; if $a<0$, it opens downward and has a maximum.

What does vertex form reveal immediately?

For $f(x)=a(x-h)^2+k$, the vertex is $(h,k)$ and the axis of symmetry is $x=h$.

What do the factors reveal in factored form?

For $f(x)=a(x-r_1)(x-r_2)$, the zeros are $x=r_1$ and $x=r_2$, giving the $x$-intercepts.

How do you find the vertex from standard form?

For $f(x)=ax^2+bx+c$, the axis of symmetry is $x=-\frac{b}{2a}$. Evaluate the function there to find the vertex's $y$-coordinate.

Find the vertex of $f(x)=x^2-6x+5$.

The vertex is $(3,-4)$, because the axis is $x=3$ and $f(3)=-4$.

What is the zero-product property?

If $uv=0$, then $u=0$ or $v=0$. This lets you solve a factored quadratic by setting each factor equal to zero.

Solve $x^2-7x+12=0$ by factoring.

Factor: $x^2-7x+12=(x-3)(x-4)$. Therefore, $x=3$ or $x=4$.

What quantity completes the square in $x^2+bx$?

For $x^2+bx$, add $\left(\frac{b}{2}\right)^2$ to complete the square. Add the same quantity to both sides of an equation.

Solve $x^2+6x-7=0$ by completing the square.

Rewrite as $(x+3)^2=16$, then use both square roots: $x+3=\pm4$. Thus, $x=1$ or $x=-7$.

What is the quadratic formula?

For $ax^2+bx+c=0$, use $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. It solves every quadratic equation in standard form.

How does the discriminant classify solutions?

The discriminant is $\Delta=b^2-4ac$. If $\Delta>0$, there are two distinct real solutions; if $\Delta=0$, one repeated real solution; if $\Delta<0$, no real solutions.