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17Calculus Precalculus - Intercepts of Polynomials

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The intercepts of a graph are the points where a graph crosses either the x-axis (x-intercepts) or the y-axis (y-intercepts). In calculus, we always work with real numbers unless explictly stated otherwise. So this page discusses only real x and y intercepts of polynomials. For intercepts of rational functions, see this page.

X-Intercepts

To determine the x-intercepts, we need to find where the graph crosses the x-axis. In these cases, the y-value is equal to zero. So in the equation, we set \(y=0\) and then solve for the x-values. X-intercepts are also called zeros or roots. We discuss this in more detail on this page.

Y-Intercepts

To determine the y-intercepts, we need to find where the graph crosses the y-axis. In these cases, the x-value is equal to zero. So in the equation, we set \(x=0\) and then solve for the y-values. If the equation represents a function (which all polynomials are), then there will be only one y-intercept since a function passes the vertical line test.

Checking Your Answer

It is always a good idea to check your answer. Doing so may result in an increase of your grade by one full letter grade. For these problems, plotting is a good way to check your answer.

Practice

Unless other instructed, determine the intercepts (both x and y) for these functions.

\( y = 2x + 1 \)

Problem Statement

Determine the x and y intercepts for \( y = 2x + 1 \).

Final Answer

x-intercept \((-1/2, 0)\)
y-intercept \((0,1)\)

Problem Statement

Determine the x and y intercepts for \( y = 2x + 1 \).

Solution

2786 video

Final Answer

x-intercept \((-1/2, 0)\)
y-intercept \((0,1)\)

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\( y = x/2 - 3 \)

Problem Statement

Determine the x and y intercepts for \( y = x/2 - 3 \).

Final Answer

x-intercept \((6, 0)\)
y-intercept \((0,-3)\)

Problem Statement

Determine the x and y intercepts for \( y = x/2 - 3 \).

Solution

2787 video

Final Answer

x-intercept \((6, 0)\)
y-intercept \((0,-3)\)

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\( 2x - 3y = 6 \)

Problem Statement

Determine the x and y intercepts for \( 2x - 3y = 6 \).

Final Answer

x-intercept \((3, 0)\)
y-intercept \((0,-2)\)

Problem Statement

Determine the x and y intercepts for \( 2x - 3y = 6 \).

Solution

2788 video

Final Answer

x-intercept \((3, 0)\)
y-intercept \((0,-2)\)

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\( 3x + 4y = 12 \)

Problem Statement

Determine the x and y intercepts for \( 3x + 4y = 12 \).

Final Answer

x-intercept \((4, 0)\)
y-intercept \((0,3)\)

Problem Statement

Determine the x and y intercepts for \( 3x + 4y = 12 \).

Solution

2789 video

Final Answer

x-intercept \((4, 0)\)
y-intercept \((0,3)\)

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\( y = x^2 + 1 \)

Problem Statement

Determine the intercepts (both x and y) for \( y = x^2 + 1 \).

Final Answer

x-intercepts: none
y-intercepts: \(y=1\)

Problem Statement

Determine the intercepts (both x and y) for \( y = x^2 + 1 \).

Solution

To find the x-intercepts, we set \(y=0\) and solve the the x-values.
\(\begin{array}{rcl} 0 & = & x^2+1 \\ -1 & = & x^2 \end{array} \)
Since there are no real values that solve this equation, we know that there are no real x-intercepts.

Now let's determine the y-intercepts. To do this, we set \(x=0\) and solve.
\( y=0^2+1 = 1\)
So, the only y-intercept is \(y=1\).

Final Answer

x-intercepts: none
y-intercepts: \(y=1\)

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\( y=2(x-1)(x+2)(x-3) \)

Problem Statement

Determine the intercepts (both x and y) for \( y=2(x-1)(x+2)(x-3) \).

Solution

2532 video

video by PatrickJMT

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\( y=x^2-3x-18 \)

Problem Statement

Determine the intercepts (both x and y) for \( y=x^2-3x-18 \).

Solution

2533 video

video by PatrickJMT

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\( f(x) = (x-2)(x+1)(x-4)(x+3) \)

Problem Statement

Determine the intercepts (both x and y) for \( f(x) = (x-2)(x+1)(x-4)(x+3) \).

Solution

2534 video

video by MIP4U

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\( f(x) = 2x^3 + 2x^2 - 24x \)

Problem Statement

Determine the intercepts (both x and y) for \( f(x) = 2x^3 + 2x^2 - 24x \).

Solution

2535 video

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\( y = -x^2 + 4x - 4 \)

Problem Statement

Determine the intercepts (both x and y) for \( y = -x^2 + 4x - 4 \).

Solution

2557 video

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