9.10.7 Practice - Algebra 1 | OpenStax

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9.10.7 Practice

9.10.7 • Practice

Complete the following questions to practice the skills you have learned in this lesson.

The following quadratic expressions all define the same function.

$(x+5)(x+3)$
$x^2+8x+15$
$(x+4)^2−1$

Select three statements that are true about the graph of this function.

The $y$ -intercept is $(0, -15)$ .
The vertex is $(-4, -1)$ .
The $x$ -intercepts are $(-5, 0)$ and $(-3, 0)$ .
The $x$ -intercepts are $(0, 5)$ and $(0, 3)$ .
The $x$ -intercept is $(0, 15)$ .
The $y$ -intercept is $(0, 15)$ .
The vertex is $(4, -1)$ .

The following expressions all define the same quadratic function.

$(x−4)(x+6)$
$x^2+2x−24$
$(x+1)^2−25$

What is the value of the $y$ -coordinate of the $y$ -intercept $(0, \_\_\_)$ of the graph of the function?

Select two $x$ -intercepts of the graph.

$(6,0)$
$(-4,0)$
$(-6,0)$
$(5,0)$
$(4,0)$

What is the $x$ -coordinate of the vertex of the graph?

What is the $y$ -coordinate of the vertex of the graph?

Is this an accurate graph of the function?

No
Yes

Here is one way an expression in standard form is rewritten into vertex form.

Original expression
$x^2−7x+6$

Step 1 - $x^2−7x+(-\frac{7}{2})^2+6−(-\frac{7}{2})^2$

Step 2 - $(x−\frac{7}{2})^2+6−\frac{49}{4}$

Step 3 - $(x−\frac{7}{2})^2+\frac{24}{4}−\frac{49}{4}$

Step 4 - $(x−\frac{7}{2})^2-\frac{25}{4}$

In Step 1, where did the number $-\frac{7}{2}$ come from?

Only the non-linear term was divided by 2.
It is half of -7 being squared.
Every term in the expression was divided by 2.
It is the substitute value for $x$ .

In Step 1, why was $(-\frac{7}{2})^2$ added and then subtracted?

It is subtracted to complete the square. It is added to change the value of the expression.
It is added to complete the square. It is subtracted to change the value of the expression.
It is subtracted to complete the square. It is added to keep the value of the expression unchanged.
It is added to complete the square. It is subtracted to keep the value of the expression unchanged.

What happened in Step 2?

The expression is added to keep the value of the expression unchanged.
The expression, which is not a perfect square, is written as a squared expression.
The expression, which is a perfect square, is written as a squared expression.
The expression is subtracted to keep the value of the expression unchanged.

What happened in Step 3?

The constant term is written as a fraction so that it has a common denominator, which makes it easier to subtract.
The linear term is written as a fraction so that it has a common denominator, which makes it easier to subtract.
A random term was added to complete the square.
The quadratic term is written as a fraction so that it has a common denominator, which makes it easier to subtract.

What does the last expression tell us about the vertex of the graph of a function defined by this expression?

The vertex of the graph is at $(\frac{5}{2}, -6)$ .
The vertex of the graph is at $(\frac{7}{2}, -6\frac{1}{4})$ .
The vertex of the graph is at $(\frac{5}{2}, -\frac{5}{2})$ .
The vertex of the graph is at $(-6\frac{1}{4}, \frac{7}{2})$ .

Enter the missing number, including the sign, for the expression when $d(x)=x^2+12x+36$ is written in vertex form.

$d(x)=(x\_\_\_)^2$ .

Enter the missing numbers, including the signs, for the expression when $f(x)=x^2+10x+21$ is written in vertex form. $f(x)=(x + \underline{\;\mathbf a\boldsymbol.\;})^2 \underline{\;\mathbf b\boldsymbol.\;}$

What is the value of the missing number, including the sign, in the place of the letter a?

What is the value of the missing number, including the sign, for the letter b?

Enter the missing numbers, including the signs, for the expression when $g(x)=2x^2−20x+32$ is written in vertex form. $g(x)= \underline{\;\mathbf a\boldsymbol.\;} (x + \underline{\;\mathbf b\boldsymbol.\;} )^2 \underline{\;\mathbf c\boldsymbol.\;}$

What is the value of the missing number, including the sign, in place of letter a?

What is the value of the missing number, including the sign, in place of letter b?

What is the value of the missing number, including the sign, in place of letter c?

Enter the missing numbers, including the signs, for the expression when $h(x)=(4x-3)^2$ is expanded to standard form. $h(x)= ( \underline{\;\mathbf a\boldsymbol.\;}\; x )^2 + 2 ( \underline{\;\mathbf b\boldsymbol.\;}\; x)(-3) + ( \underline{\;\mathbf c\boldsymbol.\;} )^2$

What is the value of the missing number, including the sign, in place of letter a?

What is the value of the missing number, including the sign, in place of letter b?

What is the value of the missing number, including the sign, in place of letter c?

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