1.2.3 Practice set: lines and quadratics

Lines

Lines (lesson 1.2.1)

0 of 9 correct

1.
Find the slope of the line through $(2, 4)$ and $(5, 10)$
2.
Find the slope of the line through $(-2, 3)$ and $(2, -5)$
3.
A tank holds 50 litres and drains at 4 litres per minute. After how many minutes is it empty?
4.
Write the line through $(1, 2)$ and $(3, 8)$ in the form $mx + c$
5.
$y = 2x + 3$ and $y = -x + 6$ meet at one point. Give its $x$
6.
Write the line parallel to $y = 4x - 1$ through $(2, 3)$ in the form $mx + c$
7.
What is the slope of $2x - 4y = 8$ ?
8.
A line has slope $-2$ and passes through $(-1, 6)$ . Find its intercept $c$
9.
A line passes through $(-2, 7)$ and $(5, 7)$ . What is its slope?

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Quadratics

Quadratics (lesson 1.2.2)

0 of 10 correct

1.
At what $x$ is the vertex of $y = x^2 - 4x + 1$ ?
2.
What is the smallest value of $x^2 - 4x + 1$ ?
3.
Complete the square: write $x^2 + 6x + 4$ as $(x + h)^2 + k$
4.
$x^2 - 2x - 8 = 0$ has two roots. What is the larger one?
5.
Find the discriminant of $x^2 + 5x + 6$
6.
Write $3x^2 - 6x + 1$ in vertex form $a(x - h)^2 + k$
7.
$2x^2 + x - 6 = 0$ has two roots. What is the larger one?
8.
At what $x$ is the vertex of $y = 2x^2 + 8x + 1$ ?
9.
What is the largest value of $y = -x^2 + 6x$ ?
10.
How many real roots does $x^2 + 2x + 5 = 0$ have?

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Coding exercise

Exercise

Vertex and roots of any quadratic

Write two functions for $ax^2 + bx + c$ , using the vertex rule and the quadratic formula from lesson 1.2.2.

vertex(a, b, c) returns the vertex as a tuple (x, y). It is the lowest point when a > 0 and the highest when a < 0.
real_roots(a, b, c) returns a list of the real roots, smallest first: two, one or none, as the discriminant says.

Both raise ValueError when a == 0, because then the expression is not a quadratic.

See the tests

def _raises_value_error(fn, *args):
    try:
        fn(*args)
    except ValueError:
        return True
    return False


def test_vertex():
    """The vertex of x² − 6x + 5 is (3, −4)"""
    expect_close(vertex(1, -6, 5), (3.0, -4.0), what="vertex(1, -6, 5)")


def test_vertex_downward():
    """The vertex of −x² + 6x is the maximum (3, 9)"""
    expect_close(vertex(-1, 6, 0), (3.0, 9.0), what="vertex(-1, 6, 0)")


def test_two_roots():
    """x² − 6x + 5 = 0 has roots 1 and 5, smallest first"""
    expect_close(real_roots(1, -6, 5), [1.0, 5.0], what="real_roots(1, -6, 5)")


def test_two_roots_negative_a():
    """−2x² + 2x + 4 = 0 has roots −1 and 2, still smallest first"""
    expect_close(real_roots(-2, 2, 4), [-1.0, 2.0], what="real_roots(-2, 2, 4)")


def test_one_root():
    """x² − 4x + 4 = 0 has the single root 2"""
    expect_close(real_roots(1, -4, 4), [2.0], what="real_roots(1, -4, 4)")


def test_no_real_roots():
    """2x² + 3x + 5 = 0 has no real roots: an empty list"""
    roots = real_roots(2, 3, 5)
    assert roots == [], f"expected [], got {roots!r}"


def test_not_a_quadratic():
    """a = 0 raises ValueError in both functions"""
    assert _raises_value_error(vertex, 0, 2, 1), "vertex(0, 2, 1) should raise ValueError"
    assert _raises_value_error(real_roots, 0, 2, 1), "real_roots(0, 2, 1) should raise ValueError"

Lines

Quadratics

Coding exercise

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