Quadratics
The graph of a quadratic crosses the x-axis twice.
What does this tell you about its discriminant?
The discriminant is positive, meaning there are two real solutions.
If f(3)=5, what point must be on the graph of f(x)?
The point is (3,5)
Why does an exponential graph never cross the x-axis?
Because exponential functions are always positive and never equal zero.
What is the midline of
y=3sin(x)+2
and what does it represent?
The midline is y=2, representing the average value of the function.
A quadratic has a positive leading coefficient and a vertex below the x-axis.
How many real solutions does it have?
Two real solutions, because the parabola opens upward and crosses the x-axis twice.
If f(x) is shifted right 4 units, what happens to the x-values in the equation?
Each x is replaced with x−4.
Between 2x and 5x, which grows faster as x increases, and why?
5x grows faster because it has the larger base.
Why does sin(x) repeat its values every 2π?
Because sine is based on the unit circle, which repeats every full rotation of 2π.
Without solving, determine whether
x2+6x+20=0
has real solutions.
No real solutions, because the discriminant
b2-4ac=36-80 is negative.
Explain why a vertical line fails the vertical line test.
A vertical line gives multiple y-values for the same x-value, so it is not a function.
Without solving, decide whether the solution to
3x=20
is closer to x=2 or x=3.
Closer to x=3, since 32=9 and 33 =20, and 20 is closer to 27.
How many solutions does
sin(x)=0
have on 0≤ x≤ 2π and why?
Three solutions: 0, π, and 2π, where the sine graph crosses the x-axis
A quadratic function has zeros at x=1 and x=7.
Is the vertex closer to x=1, closer to x=7, or exactly in the middle?
Exactly in the middle at x=4, because the vertex lies halfway between the zeros.
A function increases for all values of x.
What does this tell you about the shape of its graph?
The graph rises from left to right and never decreases.
Why are logarithms useful for solving exponential equations?
They allow you to rewrite exponents as expressions you can solve algebraically.
A cosine function has amplitude 2 and is reflected across the x-axis.
How does this change the graph?
The reflection flips the graph vertically, but the amplitude remains 2.