ISEE Middle 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

5 practice questions for ISEE Middle Level

Q1 Medium

Compare the two quantities.

Column A

The slope of the line

3 x + 6 y = 12

Column B

The slope of the line passing through

(1, - 2)

and

(5, - 4)

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

To find the quantity in Column A, rewrite the equation in slope-intercept form ( $y = m x + b$ ). Subtract $3 x$ from both sides to get $6 y = - 3 x + 12$ , then divide by 6 to get $y = - \frac{1}{2} x + 2$ . The slope is $- \frac{1}{2}$ . To find the quantity in Column B, use the slope formula $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ . Substituting the given points yields $\frac{- 4 - ( - 2 )}{5 - 1} = \frac{- 2}{4} = - \frac{1}{2}$ . The two quantities are equal.

Answer: C

Q2 Medium

What is the slope of the line

5 x - 3 y = 15

- \frac{5}{3}

- \frac{3}{5}

\frac{3}{5}

\frac{5}{3}

Show Solution

To find the slope, convert the equation into slope-intercept form ( $y = m x + b$ ), where $m$ represents the slope. First, subtract $5 x$ from both sides to get $- 3 y = - 5 x + 15$ . Next, divide every term by $- 3$ to isolate $y$ : $y = \frac{5}{3} x - 5$ . The slope of the line is $\frac{5}{3}$ .

Answer: D

Q3 Medium

A water tank is being filled at a constant rate. After 2 minutes, the tank contains 12 gallons of water. After 7 minutes, the tank contains 37 gallons of water. What is the rate at which the tank is being filled, in gallons per minute?

A 3

B 4

C 5

D 6

Show Solution

The rate at which the tank is being filled represents the slope of the linear relationship between time and water volume. Use the slope formula $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ with the points (2, 12) and (7, 37). The rate is $\frac{37 - 12}{7 - 2} = \frac{25}{5} = 5$ gallons per minute.

Answer: C

Q4 Medium

Line

k

passes through the points

(- 2, 5)

and

(2, - 3)

. What is the slope of a line that is perpendicular to line

k

- 2

- \frac{1}{2}

\frac{1}{2}

D 2

Show Solution

First, find the slope of line $k$ using the slope formula $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ . Substituting the given points gives $\frac{- 3 - 5}{2 - ( - 2 )} = \frac{- 8}{4} = - 2$ . Perpendicular lines have negative reciprocal slopes. The negative reciprocal of $- 2$ is $\frac{1}{2}$ .

Answer: C

Q5 Medium

A line has a slope of

- 3

and passes through the point

(- 2, 4)

. What is the y-intercept of the line?

- 2

B 2

C 6

D 10

Show Solution

The equation of a line in slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept. We are given the slope $m = - 3$ and a point $(x, y) = (- 2, 4)$ . Substitute these values into the equation to get $4 = - 3 (- 2) + b$ . Simplify to $4 = 6 + b$ . Subtract 6 from both sides to find $b = - 2$ . The y-intercept is $- 2$ .

Answer: A

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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