ISEE Upper Level

Linear Graphing & Slope

Q: What does f(x) actually mean?

It's just a fancy math way of writing `y`! It tells us that the output (the answer) depends on the input (the `x`). You say it out loud as 'f of x'.

Q: Do I need to memorize the slope formula for the ISEE?

Yes! The ISEE doesn't give you a formula sheet. Memorize `m = \frac{y_2 - y_1}{x_2 - x_1}` and remember to always put the `y` values on top!

Slope, y-intercept, graphing linear equations, identifying graph properties, and slope-intercept form

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Have you ever used a vending machine? You push a button (the input), and out pops a snack (the output). In math, a function is just like a magical smoothie machine! 🍓 You drop a number in, the machine chops it up, mixes it with some math rules, and pours out a brand new number.

When you see something like $f (x) = 2 x + 1$ , don't let those letters scare you! The $f$ just stands for 'function', and the $x$ is the ingredient you drop inside. If you drop in a $3$ (we write that as $f (3)$ ), the machine multiplies it by $2$ and adds $1$ to give you $7$ . Delicious! 🥤

Graphing is just a way to take a picture of what our machine is doing. We use a coordinate plane, which looks like a giant piece of grid paper. Every time we put an $x$ into our machine and get a $y$ out, we get a point (x, y). If we plot a bunch of points and connect them, we get a line!

On the ISEE, you'll see questions asking you to find points on a line, figure out the slope (how steep the line is, like a skateboard ramp! 🛹), or compare functions in the Quantitative Reasoning section. The great news? The ISEE has no penalty for guessing! So, always pick an answer, even if the math machine seems a little tricky that day. Grab your pencil and let's make some math smoothies!

Key Formula

The Magic Line Equation (Slope-Intercept Form) is

y = m x + b

. The

m

is the slope (how steep the line is) and

b

is the y-intercept (where the line crosses the y-axis). To find the slope between two points, use

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

Practice Questions

3 practice questions for ISEE Upper Level

Q1 Hard

Line

p

has the equation

2 y = 5 x - 8

. Line

q

is perpendicular to line

p

Column A

The slope of line

q

Column B

- \frac{2}{5}

A The quantity in Column A is greater.

B The quantity in Column B is greater.

C The two quantities are equal.

D The relationship cannot be determined from the information given.

Show Solution

First, find the slope of line $p$ by rewriting its equation in slope-intercept form ( $y = m x + b$ ). Divide both sides by 2 to get $y = \frac{5}{2} x - 4$ . The slope of line $p$ is $\frac{5}{2}$ . Perpendicular lines have slopes that are negative reciprocals of each other. The negative reciprocal of $\frac{5}{2}$ is $- \frac{2}{5}$ . Therefore, the slope of line $q$ is $- \frac{2}{5}$ . Since Column A is $- \frac{2}{5}$ and Column B is $- \frac{2}{5}$ , the two quantities are equal.

Answer: C

Q2 Hard

A line passes through the points

(- 2, 7)

and

(4, - 5)

. What is the y-intercept of the line?

- 2

1

3

5

Show Solution

First, find the slope ( $m$ ) of the line using the formula $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ . Substituting the given points: $m = \frac{- 5 - 7}{4 - ( - 2 )} = \frac{- 12}{6} = - 2$ . Next, use the slope-intercept form ( $y = m x + b$ ) and one of the points to find the y-intercept ( $b$ ). Using the point $(- 2, 7)$ : $7 = - 2 (- 2) + b$ , which simplifies to $7 = 4 + b$ . Subtract 4 from both sides to find $b = 3$ . The y-intercept is $3$ .

Answer: C

Q3 Hard

The equation of a line is

3 x - 4 y = 24

. At what point does the graph of the line intersect the x-axis?

(0, - 6)

B (0, 8)

C (8, 0)

D (24, 0)

Show Solution

The graph of a line intersects the x-axis at its x-intercept. The x-intercept occurs where the y-coordinate is $0$ . Substitute $y = 0$ into the given equation: $3 x - 4 (0) = 24$ . This simplifies to $3 x = 24$ . Divide both sides by 3 to get $x = 8$ . Therefore, the line intersects the x-axis at the point (8, 0).

Answer: C

Tips & Strategies

Rise over Run! When finding slope on a graph, always count how far UP or DOWN you go (rise) before you count how far LEFT or RIGHT you go (run). Putting the $x$ on top of your fraction is a super common trap!
For Quantitative Comparison questions, don't solve for $x$ or $y$ if you don't have to! If they just ask for the y-intercept, grab the $b$ from $y = m x + b$ and compare. Save your brainpower for harder questions!
Use the answers! If you forget how to graph a line, you can test the points given in the answer choices by plugging the $x$ and $y$ values back into the equation to see if it makes a true math sentence.

Common Mistakes

Watch out for negative signs when finding the slope! If you are subtracting a negative number, like $5 - (- 2)$ , remember that two negatives make a positive, so it becomes $5 + 2$ .
Don't get tricked by $f (x)$ . It does NOT mean $f$ times $x$ ! It is just the name of the function, like saying 'Machine A'.

Frequently Asked Questions

What does f(x) actually mean?

It's just a fancy math way of writing $y$ ! It tells us that the output (the answer) depends on the input (the $x$ ). You say it out loud as 'f of x'.

Do I need to memorize the slope formula for the ISEE?

Yes! The ISEE doesn't give you a formula sheet. Memorize $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ and remember to always put the $y$ values on top!

What if I get totally stuck on a function question?

Since the ISEE has no guessing penalty, never leave a question blank! If you're stuck, try plugging in simple numbers like $0$ or $1$ for $x$ to see what happens, or just make your best guess and move on.

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