Precalculus : Graph a Quadratic Function Using Intercepts, Vertex, and Axis of Symmetry

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #1 : Graph A Quadratic Function Using Intercepts, Vertex, And Axis Of Symmetry

Find the vertex, roots, and the  value that the line of symmetry falls on of the function .

Possible Answers:

vertex , the roots  and , and the axis of symmetry would fall on .

vertex , the roots  and , and the axis of symmetry would fall on .

vertex , the roots  and , and the axis of symmetry would fall on .

vertex , the roots  and , and the axis of symmetry would fall on x=.5.

vertex , the roots  and , and the axis of symmetry would fall on .

Correct answer:

vertex , the roots  and , and the axis of symmetry would fall on .

Explanation:

All quadratic functions have a vertex and many cross the x axis at points called zeros or roots. If we know the vertex and its zeros, quadratic functions become very easy to draw since the vertex is also a line of symmetry (the zeros are equidistant from the vertex on either side).

Factor the equation  to get  and . Thus, the roots are 3 and -2.

The vertex can be found by using .

simplify

.

 

The axis of symmetry is halfway between the two roots, or simply the x coordinate of the vertex. So the axis of symmetry lies on x=1/2. To graph, draw a point at the coordinate pair of the vertex. Then draw points on the x axis at the roots, and finally, trace upwards from the vertex through the roots with a gentle curve.

Example Question #2 : Graph A Quadratic Function Using Intercepts, Vertex, And Axis Of Symmetry

Which of the following functions matches the provided parabolic graph?

 

Varsity practice precalc

Possible Answers:

Correct answer:

Explanation:

Finding the vertex, intercept and axis of symmetry are crucial to finding the function that corresponds to the graph:

The vertex form of a quadratic function is written as:

and the coordinates for the vertex are:

Looking at the graph and the position of the axis of symmetry, the vertex is positioned at , leaving us with an equation so far of:

While we don't know a right away,  is the only option that really works. The y-intercept is at  and we can plug that into the formula to confirm that this is the correct function:

 

Example Question #3 : Graph A Quadratic Function Using Intercepts, Vertex, And Axis Of Symmetry

Which of the following is an equation for the parabola represented in the graph below?

Varsity practice precalc

Possible Answers:

Correct answer:

Explanation:

Immediately we can tell that the equation has a negative coefficient, because the parabola opens downward, forming an umbrella shape. Based upon the information given in the figure, we can use the intercepts, axis of symmetry, and the vertex to identify the equation of the parabola. Let's observe the vertex form of a parabola written as the following:

In this equation,  is the vertex of the parabola, and  determines whether the parabola opens upwards or downwards. The axis of symmetry is at  and the vertex is located at , which we can plug into the following function:

We know that  is negative because of the position of the parabola. 

 

 

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