SAT Math › Functions and Graphs
Find the equation of the line passing through the points and
.
To calculate a line passing through two points, we first need to calculate the slope, .
Now that we have the slope, we can plug it into our equation for a line in slope intercept form.
To solve for , we can plug in one of the points we were given. For the sake of this example, let's use
, but realize either point will give use the same answer.
Now that we have solved for b, we can plug that into our slope intercept form and produce and the answer
Find the point at which these two lines intersect:
We are looking for a point, , where these two lines intersect. While there are many ways to solve for
and
given two equations, the simplest way I see is to use the elimination method since by adding the two equations together, we can eliminate the
variable.
Dividing both sides by 7, we isolate y.
Now, we can plug y back into either equation and solve for x.
Next, we can solve for x.
Therefore, the point where these two lines intersect is .
Find the roots of the function:
Factor:
Double check by factoring:
Add together:
Therefore:
A baseball is thrown straight up with an initial speed of 50 feet per second by a man standing on the roof of a 120-foot high building. The height of the baseball in feet, as a function of time in seconds , is modeled by the function
To the nearest tenth of a second, how long does it take for the baseball to hit the ground?
When the baseball hits the ground, the height is 0, so we set . and solve for
.
This can be done using the quadratic formula:
Set :
One possible solution:
We throw this out, since time must be positive.
The other:
This solution, we keep. The baseball hits the ground in about 4.7 seconds.
Find the y-intercept of the following line.
To find the y-intercept of any line, we must get the equation into the form
where m is the slope and b is the y-intercept.
To manipulate our equation into this form, we must solve for y. First, we must move the x term to the right side of our equation by subtracting it from both sides.
To isolate y, we now must divide each side by 3.
Now that our equation is in the desired form, our y-intercept is simply
What is the domain of the following function? Please use interval notation.
A basic knowledge of absolute value and its functions is valuable for this problem. However, if you do not know what the typical shape of an absoluate value function looks like, one can always plug in values and plot points.
Upon doing so, we learn that the -values (domain) are not restricted on either end of the function, creating a domain of negative infinity to postive infinity.
If we plug in -100000 for , we get 100000 for
.
If we plug in 100000 for , we get 100000 for
.
Additionally, if we plug in any value for , we will see that we always get a real, defined value for
.
**Extra Note: Due to the absolute value notation, the negative (-) next to the is not important, in that it will always be made positive by the absolute value, making this function the same as
. If the negative (-) was outside of the absolute value, this would flip the function, making all corresponding
-values negative. However, this knowledge is most important for range, rather than domain.
Red line
Blue line
Green line
Purple line
None of them
A parabola is one example of a quadratic function, regardless of whether it points upwards or downwards.
The red line represents a quadratic function and will have a formula similar to .
The blue line represents a linear function and will have a formula similar to .
The green line represents an exponential function and will have a formula similar to .
The purple line represents an absolute value function and will have a formula similar to .
Consider the equation:
The vertex of this parabolic function would be located at:
For any parabola, the general equation is
, and the x-coordinate of its vertex is given by
.
For the given problem, the x-coordinate is
.
To find the y-coordinate, plug into the original equation:
Therefore the vertex is at .
Solve the following function:
and
You must get by itself so you must add
to both side which results in
.
You must get the square root of both side to undue the exponent.
This leaves you with .
But since you square the in the equation, the original value you plug can also be its negative value since squaring it will make it positive anyway.
This means your answer can be or
.
Solve:
Rewrite the right side as base 2.
Replace the term into the equation.
With similar bases, we can set the exponents equal.
Subtract six from both sides.
Divide by negative three on both sides.
The answer is: