Task Sets: Graph linear and quadratic functions - Common Core Math - HS Functions

A

Recall / Warm-Up

1.

What is the slope of the line $f(x) = -3x + 7$ ?

A.

7

B.

-3

C.

3

D.

-7

2.

What is the $y$ -intercept of the line $3x + 4y = 12$ ?

A.

$(4, 0)$

B.

$(0, 3)$

C.

$(0, 4)$

D.

$(3, 0)$

3.

For a quadratic function $f(x) = ax^2 + bx + c$ , which condition tells you the graph opens downward and the vertex is a maximum?

A.

$a > 0$

B.

$a < 0$

C.

$c > 0$

D.

$b > 0$

B

Fluency Practice

1.

What is the $x$ -intercept of the line $f(x) = 4x - 8$ ?

A.

$(0, -8)$

B.

$(2, 0)$

C.

$(-8, 0)$

D.

$(8, 0)$

2.

The line $y - 5 = 3(x - 2)$ is in point-slope form. Which of the following correctly identifies the slope and a point on the line?

A.

Slope $= 3$ , passes through $(2, 5)$

B.

Slope $= 3$ , passes through $(-2, -5)$

C.

Slope $= -3$ , passes through $(2, 5)$

D.

Slope $= 5$ , passes through $(2, 3)$

3.

For $f(x) = x^2 - 6x + 5$ , find the following without graphing:
(a) the $y$ -intercept,
(b) the axis of symmetry,
(c) the vertex,
(d) the $x$ -intercepts (if any).
State whether the vertex is a maximum or minimum.

4.

For $g(x) = -2x^2 + 8x - 3$ , find:
(a) the axis of symmetry,
(b) the vertex,
(c) the $y$ -intercept.
Does the graph have a maximum or minimum? What is that value?

5.

For $h(x) = x^2 + 2x + 5$ , compute the discriminant. How many $x$ -intercepts does the graph have? Find the vertex and $y$ -intercept, and describe the graph.

C

Varied Practice

D

Word Problems

1.

A company's profit function is modeled by $P(x) = 2(x - 10)(x - 80)$ dollars, where $x$ is the number of items sold.

Find the $x$ -intercepts of $P$ and explain what they represent in context. Then find the axis of symmetry and the vertex.

2.

A ball is launched upward. Its height in feet is given by $h(t) = -16t^2 + 64t + 5$ , where $t$ is time in seconds.

1.

Does the ball reach a maximum height or a minimum height? How do you know?

A.

Maximum height, because the leading coefficient $-16$ is negative.

B.

Minimum height, because the leading coefficient $-16$ is negative.

C.

Maximum height, because the constant term 5 is positive.

D.

Minimum height, because the ball starts at height 5 and goes up.

2.

Find the maximum height and the time at which it occurs.

3.

A company models its revenue with $R(x) = -0.5x^2 + 30x$ , where $x$ is the number of units sold and $R$ is in dollars.

Find the number of units that maximizes revenue and state the maximum revenue. Include units in your answer.

E

Error Analysis

1.

A student is asked to graph $f(x) = 2(x - 3)^2 + 1$ and writes: "Vertex is at the origin $(0, 0)$ because the parent function $y = x^2$ has its vertex there. The function just shifts 3 right and 1 up, but I still draw the vertex at $(0, 0)$ ."

What is the student's error? What is the actual vertex?

A.

The student correctly identified the shifts but placed the vertex at the starting point. The vertex after shifting 3 right and 1 up is $(3, 1)$ .

B.

The student has the shifts backward. The vertex is $(-3, -1)$ .

C.

The student has the vertex correct. The vertex is always at the origin for parabolas.

D.

The vertex is at $(2, 0)$ because the coefficient 2 moves the vertex horizontally.

2.

A student finds the intercepts of $f(x) = 3x - 9$ . The student writes:
" $x$ -intercept: plug in $x = 0$ . $f(0) = 3(0) - 9 = -9$ . So the $x$ -intercept is $(0, -9)$ .
$y$ -intercept: set $f(x) = 0$ . $0 = 3y - 9$ . Hmm, that doesn't make sense."

What mistake did the student make, and what are the correct intercepts?

A.

The student switched the procedures. For the $x$ -intercept, set $f(x) = 0$ and solve for $x$ ; for the $y$ -intercept, plug in $x = 0$ . The correct intercepts are $(3, 0)$ and $(0, -9)$ .

B.

The student made an arithmetic error. The $x$ -intercept is $(0, -9)$ but the $y$ -intercept is $(0, 9)$ .

C.

Both procedures are correct; the student just needs to simplify the second equation.

D.

The student should have found the slope first, then the intercepts follow automatically.

F

Challenge / Extension

1.

A parabola has vertex $(2, -3)$ and passes through the point $(5, 6)$ . Write the equation of the parabola in vertex form $y = a(x - h)^2 + k$ . Show all steps.

2.