Exponential and trig graphs Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Exponential and trig graphs questions. See exactly how to solve problems on exponential graph, y-intercept, k^x, asymptote.

exponential graphy-interceptk^xasymptotefinding ksubstitution
GCSE Higher70 questionsStep-by-step solutions
Question 1
1 markeasy
The curve y=3xy = 3^x is drawn for −2≤x≤3-2 \le x \le 3. Write down the coordinates of the point where the curve crosses the yy-axis.

Worked solution

  1. Use the y-axis

    x=0x = 0

    Every point on the y-axis has x=0x = 0, so substitute x=0x = 0 into the equation.

  2. Substitute x=0x = 0

    y=30=1y = 3^0 = 1

    Any positive number raised to the power 0 is 1.

  3. Write the coordinates

    (0, 1)(0,\ 1)

    The curve crosses the y-axis at the point (0, 1).

Answer
(0, 1)(0,\ 1)
Question 2
1 markeasy
Write down the equation of the asymptote of the curve y=5xy = 5^x.

Worked solution

  1. Think about large negative x

    5−10=15105^{-10} = \frac{1}{5^{10}}

    As x becomes more and more negative, 5x5^x becomes a very small positive number.

  2. Decide the limit

    5x→0 as x→−∞5^x \to 0 \text{ as } x \to -\infty

    The curve gets closer and closer to the x-axis but never reaches it.

  3. State the asymptote

    y=0y = 0

    The x-axis is the horizontal asymptote, and its equation is y=0y = 0.

Answer
y=0y = 0
Question 3
1 markeasy
The curve y=kxy = k^x passes through the point (1,6)(1, 6). Write down the value of kk.

Worked solution

  1. Substitute the point

    6=k16 = k^1

    The point (1, 6) lies on the curve, so x=1x = 1 gives y=6y = 6.

  2. Simplify the power

    k1=kk^1 = k

    Anything to the power 1 is itself.

  3. State k

    k=6k = 6

    So the curve is y=6xy = 6^x.

Answer
k=6k = 6
Question 4
1 markeasy
The curve y=2xy = 2^x is drawn. Work out the value of yy when x=4x = 4.

Worked solution

  1. Substitute x=4x = 4

    y=24y = 2^4

    Replace x with 4 in y=2xy = 2^x.

  2. Work out the power

    24=2×2×2×22^4 = 2 \times 2 \times 2 \times 2

    Multiply four 2s together.

  3. State the value

    y=16y = 16

    So the curve passes through (4, 16).

Answer
y=16y = 16
Question 5
2 markseasy
The curve y=10xy = 10^x is drawn. Work out the value of yy when x=−1x = -1.

Worked solution

  1. Substitute x=−1x = -1

    y=10−1y = 10^{-1}

    Replace x with -1 in y=10xy = 10^x.

  2. Use the negative index law

    10−1=110110^{-1} = \frac{1}{10^{1}}

    A negative index means the reciprocal of the positive power.

  3. State the value

    y=0.1y = 0.1

    So the curve passes through (-1, 0.1), just above the x-axis.

Answer
y=0.1y = 0.1

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