Algebraic Foundations
What is the value of 3(2x + 1) when x = 4?
27
Is the relation {(1,2), (2,3), (3,4)} a function? Explain why.
Yes, it is a function—each input (x-value) maps to exactly one output (y-value).
Sarah earns $12 per hour. Write an expression for her earnings after h hours.
12h
What is wrong with this equation: 3(x+2)=3x+2
Incorrect distribution: 3(x+2) should be 3x+6, not 3x+2.
What is one way to check your answer after solving an equation?
Plug your solution back into the original equation to verify it makes both sides equal.
Simplify: 4x−2(3x−5)
-2x+10
What is the slope of the line passing through (1, 3) and (3, 7)?
2
A rectangular garden has a length 3 meters more than its width. If the area is 40 m², write and solve the equation.
x = 5, length = 8
Explain why the equation x2+4=−1 has no real solution.
No real solution: x2+4=−1⇒x2=−5. Square roots of negatives are imaginary.
If a question looks hard, what is a good test strategy to use?
Skip it and come back later; answer easier questions first to build confidence.
Factor completely: x2−9
(x+3)(x-3)
Given f(x)= 2x2-4x, find f(3)
6
A car’s value decreases by $2,000 per year. Model the value V, after t years if it starts at $20,000.
V=20000−2000t
Justify each step in solving: 2x+4=12.
Step 1: Subtract 4 → 2x=8 Step 2: Divide by 2 → x = 4x
Why is boxing/marking key words helpful in word problems?
Helps identify important operations, quantities, or what the question is really asking.
Solve for x: 2x+5=3x−7
x=12
Compare the graphs of y=x2 and y=x2+3. How are they different?
y=x2+3 is a vertical shift up 3 units from y=x2
You are twice as old as your sister. The sum of your ages is 27. Write and solve an equation to find your age.
(x/2)(x)+(x/2)=27⇒ (3x/2)=27 ⇒x=18
Why does the graph of y=x2 always open upwards?
Because the coefficient of x2 is positive (1), which causes a U-shaped graph opening upward.
How can graphing help you verify solutions to equations?
Graphing helps visualize where equations cross the x-axis (solutions/roots).
Rewrite the expression (2x2-8)/4x in simplest form
(x-2)(x+2)/2x
True or False:
A function is defined as f(x) =
True
A company's profit P is given by P=−2x2+8x+10. What number of units x gives the maximum profit?
x = (-b/2a); 2 units
Show that (x+2)2−4 and x2+4x+ are not equivalent.
Expand (x+2)2−4=x2+4x+4−4=x2+4x. So, now they are equivalent—this was a trick! They are equivalent. ✔️
If you finish the test early, what should you do with the remaining time?
Use that time to double-check calculations, reread word problems, and verify tricky answers.