Evaluating Quadratic Expressions Using Order of Operations

Hello, and welcome to the first lesson in your graphing-functions course. Over the next few days, you will build from the algebra needed to graph parabolas accurately into recognizing and sketching cubic, reciprocal, and exponential functions.

You already know how coordinates work and how an input and output form a point on a graph. This lesson adds the key algebraic habit behind every value table for a parabola: evaluate a quadratic rule carefully at a chosen input. By the end, you should be able to turn an instruction such as “find $f(-3)$” into a correct output and a coordinate on the graph.

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A function rule is an input-output instruction

A quadratic function has an $x^2$ term. In general, it can be written as

$$f(x)=a{x}^2+bx+c$$

where $a$, $b$, and $c$ are numbers and $a\ne 0$.

For now, view this formula as a rule or machine:

• The number inside the parentheses in $f()$ is the input.
• Applying the rule produces an output.
• That input-output pair becomes a graph point: $(x,f(x))$.

For example, if

$$f(x)=x^2-2x+11$$

then $f(-3)$ asks:

When the input is $-3$, what output does the rule produce?

The central procedure is always the same:

1. Replace every occurrence of the variable with the given input.
2. Put a negative input in parentheses.
3. Simplify using the order of operations: exponents, multiplication, then addition and subtraction.

The parentheses are not cosmetic. They preserve the meaning of a negative input when it is squared.

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Substitute first; calculate second

A reliable written layout prevents most errors. Suppose

$$g(x)=3x^2+4x-5$$

and we need $g(2)$. Replace every $x$ with $2$:

$$\begin{array}{rl}g(2)& =3(2{)}^2+4(2)-5\\ & =3(4)+8-5\\ & =12+8-5\\ & =15\end{array}$$

Thus,

$$g(2)=15$$

and the corresponding point on the graph is

$$(2,15)$$

Notice that $3(2{)}^2$ does not mean $(3\cdot 2{)}^2$. The exponent belongs only to the input $2$, so square first and then multiply by $3$.

Function notation and $y$-notation

You may see the same rule written either way:

$$y=x^2+5$$

or

$$f(x)=x^2+5$$

They express the same input-output relationship. If $x=-2$, then evaluating $f(-2)$ is the same as finding the value of $y$ when $x=-2$. In both notations, the graph point is $(-2,y)$, or equivalently $(-2,f(-2))$.

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Negative inputs: parentheses protect the sign

Negative inputs are the most common source of errors in early quadratic work. Compare the following two expressions:

$$(-3{)}^2=9$$

but

$$-3^2=-9$$

In the second expression, the exponent applies to $3$ before the negative sign is applied. When the input itself is $-3$, however, we need the first expression: $(-3{)}^2$.

Consider

$$h(x)=2x^2-3x+1$$

Evaluate $h(-2)$.

Step 1: Substitute with parentheses.

$$h(-2)=2(-2{)}^2-3(-2)+1$$

Step 2: Evaluate the exponent.

$$h(-2)=2(4)-3(-2)+1$$

Step 3: Multiply, including the signs.

$$h(-2)=8+6+1$$

Step 4: Add.

$$h(-2)=15$$

So the point $(-2,15)$ lies on the graph of $h$.

Two details made this work:

• Squaring $-2$ gives a positive result: $(-2{)}^2=4$.
• The term $-3(-2)$ is positive because subtracting a negative product gives addition.

A useful checking habit is to keep the substituted expression intact for one line before doing arithmetic. Do not try to substitute and simplify mentally in one jump.

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Zero is an especially informative input

The input $0$ is quick to evaluate because every term containing $x$ becomes $0$. For

$$p(x)=4x^2-7x+6$$

we have

$$\begin{array}{rl}p(0)& =4(0{)}^2-7(0)+6\\ & =0-0+6\\ & =6\end{array}$$

Therefore,

$$p(0)=6$$

This gives the point

$$(0,6)$$

Because points with $x=0$ lie on the vertical axis, $p(0)$ is the $y$-intercept. Later, when you sketch full parabolas, evaluating at zero will give you one important graph feature immediately.

More generally, for

$$f(x)=a{x}^2+bx+c$$ $$f(0)=c$$

The constant term $c$ is the output when the input is zero.

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From evaluations to a table of values

A graph is made from many input-output pairs. A table of values organizes those calculations so that you can plot the resulting coordinates.

For the rule

$$y=x^2+5$$

take the inputs $-2,-1,0,1,2$. Each time, square the input and then add $5$.

Input $x$	$x^2$	$y=x^2+5$	Coordinate
$-2$	$4$	$9$	$(-2,9)$
$-1$	$1$	$6$	$(-1,6)$
$0$	$0$	$5$	$(0,5)$
$1$	$1$	$6$	$(1,6)$
$2$	$4$	$9$	$(2,9)$

The repeated outputs are meaningful: opposite inputs such as $-2$ and $2$ have the same square. This pattern will become the left-right symmetry of the basic parabola.

Use the worksheet’s first table as a visual model for the completed table above. Then apply the same routine to its second function, $y=x^2-10$: make the square row first, then apply the $-10$, then write each coordinate. At this stage, the goal is not a polished curve; it is to make every output trustworthy before plotting.

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A quick error-checking routine

When you evaluate a quadratic at a number, run this short audit before moving on:

1. Did I replace every $x$?
   In $2x^2-3x+1$, there are two occurrences of $x$, so both must receive the input.

2. Did I use parentheses around a negative input?
   Write $(-4{)}^2$, not $-4^2$, when the input is $-4$.

3. Did I square before multiplying or adding?
   In $5(-2{)}^2+3$, calculate the square first.

4. Did I preserve the sign of each term?
   For example, $-2(-3)=6$, while $2(-3)=-6$.

5. Did I state the result in the requested form?
   $f(2)=15$ is a function value; $(2,15)$ is the corresponding graph point. They communicate related, but different, things.

This is the exact algebraic foundation needed for plotting: choose $x$-values, evaluate the rule, record coordinates, and place those points on a grid.

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Key takeaways

Evaluating a quadratic function means finding its output for a specified input. The disciplined method is:

• substitute the input for every variable,
• use parentheses for negative inputs,
• follow the order of operations,
• interpret the result as a graph point $(x,f(x))$.

In particular, $f(0)$ gives the $y$-intercept, and a table of evaluations supplies the points needed to graph a parabola.

Next, you will use this process to create a full value table for the parent function

$$y=x^2$$

and plot its characteristic U-shaped graph.